Symplectic Groups, Symplectic Spreads, Codes, and Unimodular Lattices

194, 113]156 Ž1997. JA966902

JOURNAL OF ALGEBRA ARTICLE NO.

Symplectic Groups, Symplectic Spreads, Codes, and Unimodular Lattices Rudolf Scharlau* Department of Mathematics, Uni¨ ersity of Dortmund, 44221 Dortmund, Germany

and Pham Huu Tiep† Department of Mathematics, Ohio State Uni¨ ersity, Columbus, OH 43210 Communicated by Walter Feit Received April 2, 1996; revised June 1996

It is known that the symplectic group Sp 2 nŽ p . has two Žcomplex conjugate. irreducible representations of degree Ž p n q 1.r2 realized over QŽ y p ., provided that p ' 3 mod 4. In the paper we give an explicit construction of an odd unimodular Sp 2 nŽ p . ? 2-invariant lattice DŽ p, n. in dimension p n q 1 for any p n ' 3 mod 4. Such a lattice has been constructed by R. Bacher and B. B. Venkov in the case p n s 27. A second main result says that these lattices are essentially unique. We show that for n G 3 the minimum of DŽ p, n. is at least Ž p q 1.r2 and at most p Ž ny1.r2 . The interrelation between these lattices, symplectic spreads of Fp2 n , and self-dual codes over Fp is also investigated. In particular, using new results of U. Dempwolff and L. Bader, W. M. Kantor, and G. Lunardon, we come to three extremal self-dual ternary codes of length 28. Q 1997 Academic Press

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1. INTRODUCTION Let p be an odd prime, and set S s Sp 2 nŽ p . for the symplectic group over Fp . In w14x, Gow considered Euclidean integral lattices in the space of the Weil representation of S. More precisely, the Weil representation W of S is a complex representation of degree p n that can be obtained from 1q 2 n Ž the action of S on the extraspecial group pq as the outer automor*E-mail: [email protected]. † E-mail: [email protected]. Supported in part by the Alexander von Humboldt Foundation. 113 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.

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phism group.. See, for example, w18, 27, or 32x for a more general approach. W is a sum of two irreducible representations that have degrees Ž p n y 1.r2 and Ž p n q 1.r2. ŽIt seems that these characters were first treated in w5x.. One of these representations, which we shall denote by W1 , is faithful and has even degree, and the kernel of the other representation, W 2 , is just the centre ZŽ S . , C2 of S. Following w14x, we shall refer to W 1 and W 2 as Weil representations. Suppose now that p ' 3 mod 4. It is shown in w14x that the characters c i of the W i , 1 F i F 2, each generate the field QŽ yp . and have Schur index 1 over Q. Hence, there exists an absolutely irreducible QG-module V that affords the S-character c q c , where c s c 1 or c 2 , and G s Sp 2 nŽ p . ? 2 s C2 ? Aut Ž S .. More precisely, if W denotes the natural 2 ndimensional S-module over Fp , with the symplectic form ² ? , ? :, then

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G s  w g GL Ž W . < 'k s "1 ; u, ¨ g W ² w Ž u . , w Ž ¨ . : s k ² u, ¨ : 4 . Thus, G is generated by S and some involution q , which induces an outer automorphism of order 2 of S. The main result of w14x is that when n is even and c s c 1 , every ZG-lattice in V Žafter suitably rescaling the inner product. is even and unimodular. ŽHere we would like to draw the reader’s attention to the following circumstance. There exist just two double extensions of type H s S ? 2 of S with ZŽ H . s ZŽ S . s CH Ž S ., which are isoclinic to each other. Therefore, one should always specify, which double extension of S we are working with. For more details on isoclinic groups cf. Lemma 2.11 w29x.. The same lattices have been considered by B. H. Gross w15x in the context of the so-called globally irreducible representations Žfor the precise definition see w15x, w29x.. When p ' 3 mod 4 and c Ž1. s Ž p n y 1.r2, the representation V is globally irreducible. Gross also shows that, for p odd, there are two globally irreducible representations V of Sp 2 nŽ p 2 . of dimension p 2 n y 1 over Q, which are related to the Weil representations and lead to even unimodular Euclidean lattices Žof dimension p 2 n y 1.. A model for these lattices when S s SL 2 Ž p 2 . is due to Elkies Žsee w15x.. Recently, Dummigan w10x has realized these even unimodular lattices as sublattices of the Mordell]Weil lattice of certain elliptic curves. He also establishes that the minimum min L s min  Ž ¨ , ¨ . < ¨ g L _  0 4 4 of such lattices L is at least Ž p n q 1.r2. On the other hand, it was proved in w29, Theorem 5.1, Corollary 5.2x, that if p ' 1 mod 4, then some exten˜ , Sp 2 nŽ p . ? 2 has a globally irreducible representation of dimension G sion 2Ž p n y 1., which affords the S-character 2Ž c 1 q c 1U .. Here ) denotes the algebraic conjugation of the field QŽ p .. Moreover, this representation also gives us even unimodular root-free lattices.

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In this paper, we are concerned with the Weil representation W 1 of the symplectic group S s Sp2 n Ž p . with the character c s c 1 of degree Ž p n q 1.r2, where p ' 3 mod 4 and n is odd. Under these assumptions, this representation is not globally irreducible; namely, c mod 2 s 1 S q h for some h g IBr 2 Ž S .. ŽWe mention that all globally irreducible representations of S have been classified in w29, 30x for the case n F 2.. Nevertheless, it also leads to unimodular lattices Ževen unimodular lattices, if p ' y1 mod 8., of dimension p n q 1. This phenomenon has first been observed by Gow w14x for the case n s 1, and by Bacher and Venkov w1x for the case p n s 27. They have also given an explicit construction of such a lattice Žin the case p n s 27. and conjectured the existence for other p n ' 3 mod 4. As it is shown in w14x, the group G s Sp 2 nŽ p . ? 2 possesses an absolutely irreducible QG-module V of dimension p n q 1 which affords the S-character c q c . If is clear that there exists a unique Žup to scalar. G-invariant positive definite symmetric bilinear form Ž?, ? . on V. In Section 2 we shall prove the following. THEOREM 1.1. Under the abo¨ e assumptions and notation, V contains a G-in¨ ariant unimodular lattice. If in addition p n ' y1 mod 8, then V contains an S-in¨ ariant e¨ en unimodular lattice. If n ) 1, these lattices ha¨ e no roots. One basic idea in the proof is to derive restrictions on the local behaviour Žthe genus, or equivalently, the discriminant group. of an arbitrary invariant lattice in the representation space corresponding to some character l of a finite group from the reduction l mod p. In Section 3, we shall give an explicit construction of a unimodular G-invariant lattice in dimension p n q 1 for any p, n with p n ' 3 mod 4. Actually, the following result, to be proved in Section 5, shows that V contains precisely one G-invariant unimodular lattice D s DŽ p, n.. Let D0 denote the sublattice of index 2 in D consisting of all vectors of even norm in D. Furthermore, set D 1 s 2Ž D0 .*, two times the dual lattice of D0 . Then D0 and D 1 n are G-invariant integral lattices, of determinant 2 2 and 2 2 p , respectively. ŽObserve that we can rescale D 1 by 1r2 such that it becomes integral of n determinant 2 p y1 .. Two integral lattices Ž G, Ž?, ? .. and Ž G9, Ž?, ? .9. are called similar if there exists a surjective homomorphism f : G ª G9 and a scalar l g Q such that Ž f Ž u., f Ž ¨ ..9 s lŽ u, ¨ . for any u, ¨ g G. THEOREM 1.2. Assume that p n ' 3 mod 4 and that G s Sp 2 nŽ p . ? 2 acts faithfully and irreducibly on an integral lattice G of rank p n q 1. Then G is similar to one of the lattices D, D0 , D 1. We recall that the full automorphism groups of all G-invariant lattices L in V have been determined in w28x. In particular, if n G 3 then either

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AutŽ L . s G, or p s 3 and AutŽ L . s Ž C3 = S . ? C2 . In Section 5, we give also an explicit construction of Žsome. Sp 2 nŽ p .-invariant even unimodular lattices in V. In Section 6, we show that if n G 3 then the minimum min D of the lattice D s DŽ p, n. is at least Ž p q 1.r2 and at most p Ž ny1.r2 . In Section 4, the case n s 3 is considered in more detail. We investigate the interrelation between our lattices, symplectic spreads of Fp2 n, and self-dual codes over Fp . In particular, three extremal self-dual ternary codes of length 28 are exhibited. The 2 ? PSp 2 nŽ p . ? 2-invariant Euclidean lattices in dimension p n q 1, provided that p n ' 1 mod 4, will be investigated in a subsequent paper w26x.

2. THE 2-PART OF THE DISCRIMINANT GROUP OF A LATTICE Throughout this section, G is an arbitrary finite group, L a G-invariant integral Euclidean lattice. The discriminant group L*rL of L is an abelian group of order det L. We make use of the canonical decomposition L*rL s O 2 Ž L*rL . [ O 2 9Ž L*rL ., where O 2 Ž ??? . denotes the 2-part and O 2 9Ž ??? . the 2-prime-part of an abelian group. The following lemma is well known and has been used in various contexts; see, for instance, w13, 31, 25x. The method essentially goes back at least to Watson w34x. LEMMA 2.1. For any integral lattice L, there exists an AutŽ L .-in¨ ariant integral o¨ er-lattice G = L such that O 2 Ž G*rG . is elementary abelian and O 2X Ž G*rG . , O 2 9Ž L*rL .. The proof is by induction on m, where 2 m ? l with odd l is the exponent of L*rL. One considers the lattice L q 2 my 1 lL*. For lattices without groups, the following lemma is a classical lemma of Kneser w19x. The extension to G-lattices is immediate. In view of its importance for the rest of the paper, we nevertheless give a proof here. LEMMA 2.2. Let G be a group without factor groups C3 or S 3 . Assume that G admits an integral lattice L of determinant 4 l for some odd integer l. Then G admits an integral lattice D with L ; D ; L* and D*rD , O 2 9Ž L*rL .. Proof. First observe that the desired D has to be an over-lattice of index 2, and any such integral over-lattice will satisfy our claim. If A s O 2 Ž L*rL . is isomorphic to the cyclic group C4 , the existence of D follows from Lemma 2.1. So let us suppose that A ( C2 = C2 . The lattices D are in one-to-one correspondence with those of the three subgroups ² ¨ : s

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² ¨ q L : ; A of order 2 which are isotropic, q Ž ¨ . s 0, with respect to the discriminant form q: A ª 12 ZrZ, q Ž ¨ . s Ž ¨ , ¨ . q Z. There are two cases. If q is not the zero map, then, since q is F 2-linear, there exists a unique subgroup ² ¨ : ; A with q Ž ¨ . s 0. It is necessarily left invariant by the whole group AutŽ L .. If q is identically zero, all three subgroups ² ¨ : will do. If none of them were invariant under G, then G would permute them transitively, and thus posses a factor group C3 or Ý3 , contradicting our assumption. Remark 2.3. One has no odd analogue of Lemma 2.2 as it is shown by the following example. Let L s ² e1 , e2 , e3 , e4 : Z be a lattice with the Gram matrix diagŽ1, 1, p, p ., where p ' 3 mod 4. Then pL* ; L ; L*, and the quadratic form u q L ¬ pŽ u, u. mod p defined on L*rL s Fp2 , has Witt index 0, as y1 is not a square modulo p. As a consequence, no unimodular Žintegral. lattice G can be isometrically embedded between L and L* Ži.e., L ; G ; L*.. This explains the special role of the prime p s 2 in the following propositions. If V is a vector space over some field F together with a nondegenerate symmetric bilinear form, we denote by det V g F rF 2 its determinant, i.e., the square class of the determinant of the Gram matrix with respect to some basis of V. v

v

PROPOSITION 2.4. Assume that a finite group G acts with character x of e¨ en degree on a quadratic ¨ ector space V o¨ er Q. Assume in addition that Ži. For e¨ ery odd prime p, x mod p is either an absolutely irreducible Brauer character, or a sum r q r , where r g IBr p Ž G . is not of quadratic type; Žii. Either x mod 2 s 2 ? 1 G q h for some h g IBr 2 Ž G ., or x mod 2 s 2 ? 1 G q h q h , where h g IBr 2 Ž G . is not self-dual; Žiii. det V / 2Q 2 . v

Then the space V contains a G-in¨ ariant unimodular lattice. Proof. Consider a G-invariant integral lattice G on V with minimal determinant. Then G is not integral for any rescaled form Ž1rs .Ž?, ? . with s g Z and s G 2. Suppose that det G ) 1. First we claim that det G s 2 m for some natural m. For, suppose that an odd prime p divides det G. Then consider the form Ž x, y . p s Ž x, y . mod p on W s GrpG, where x s x q pG, y s y q pG. As p divides det G, this G-invariant symmetric bilinear form is degenerate, and so its kernel Ž G l pG*.rpG is nonzero, i.e., G = G l pG* > pG. If G s G l pG*, then one could rescale by 1rp, a contradiction. Therefore GrŽ G l pG*. is a nonzero G-module, which supports a nondegenerate G-invariant symmet-

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ric bilinear form, Ž?, ? . p . This means that x s r q r , where r is of quadratic type, contradicting condition Ž i .. Now applying Lemma 2.1 one can suppose that G = 2 G* = 2 G. Arguing as above, we see that W s Gr2G* is a nonzero G-module, which supports a nondegenerate G-invariant symmetric bilinear form, Ž?, ? . 2 . Beside that, dim W - dim G. It is clear that Ž?, ? . 2 allows us to identify W with the dual module W *, and so the G-character of W must be real-valued. We have to distinguish the following subcases: Ža. W affords the G-character 1 G or x mod 2 y 1 G . Then G*rG , 2 G*r2G affords the G-character x mod 2 y 1 G or 1 G , respectively. In particular, det G s Ž G* : G . is equal to 2 deg xy1 or 2, which contradicts Žiii.. Žb. W affords the G-character x mod 2 y 2 ? 1 G . Then det G s 4, and one can apply Lemma 2.2. ŽHere we do not need the absence of normal subgroups of index 3 or 6 in G, since condition Žii. on x mod 2 implies that the kernel in the proof of Lemma 2.2 has index 1 or 2 in G.. Žc. W affords the G-character 2 ? 1 G . Then Ž G : 2 G*. s 4. Taking G9 s '2 G* Žwhich is equivalent to considering the same module G* but rescaling the scalar product by the constant 2., one sees that G9 is a G-invariant integral lattice and det G9 s Ž G9* : G9 . s

1

ž'

2

G : '2 G* s Ž G : 2 G* . s 4.

/

Now apply Lemma 2.2 to G9. We now give a slight extension of Proposition 2.4. PROPOSITION 2.5. Let G be a finite perfect group: G s G9. Assume that G acts on a rational quadratic ¨ ector space with character x of e¨ en degree. Assume in addition that Ži. For e¨ ery odd prime p, x mod p cannot be written as a sum a q b of some Ž nonzero. Brauer characters a , b with properties: a is of quadratic type and a can be written o¨ er Fp ; Žii. Either x mod 2 s k ? 1 G q h for some h g IBr 2 Ž G ., k g N; or x mod 2 s k ? 1 G q h q h with h g IBr 2 Ž G . not self-dual and k g N; Žiii. det V / 2Q 2 . v

Then V contains a G-in¨ ariant unimodular lattice. Proof. Using the same arguments as in the proof of Proposition 2.4, we obtain a lattice on V with det G s 2 m for some natural m and G = 2 G* = 2 G. Moreover, W s Gr2G* is a nonzero G-module which supports a nondegenerate G-invariant symmetric bilinear form Ž?, ? . 2 . Besides this,

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dim W - dim G. As the G-character of W is real-valued, we have to distinguish the following three subcases. Ža. W affords the G-character l ? 1 G or x mod 2 y l ? 1 G with l odd. Then G*rG , 2 G*r2G affords the G-character x mod 2 y l ? 1 G or l ? 1 G , respectively. In particular, det G s Ž G* : G . is equal to 2 deg xyl or 2 l , i.e., det V s 2Q 2 , a contradiction. Žb. W affords the G-character x mod 2 y l ? 1 G with l e¨ en. Then consider the F 2 G-module T s G*rG with character l ? 1 G . Our group G acts naturally on T, with kernel K say. Furthermore, the factor group GrK is embedded into some upper-triangular subgroup of SL l Ž2.. In particular, GrK is solvable. But G s G9 by our assumption, therefore K s G; i.e., G acts trivially on T. This means that the inverse image in G* of any subspace of T is G-invariant. Now endow T with the discriminant bilinear form bŽ u, ¨ . [ 2Ž u, ¨ . mod 2, and set q Ž u. s bŽ u, u.. Notice that b is well-defined and nondegenerate. First we suppose that q s 0. Then b is alternating. The inverse image of a maximal totally isotropic subspace T is the desired G. Now we consider the case where q / 0. Then A s Ker q is a F 2-space of odd dimension l y 1. Set AH s  x g U < bŽ x, A. s 04 . If A l AH s 0, then b < A is a nondegenerate alternating bilinear form on A, contradicting the fact that dim A is odd. Therefore, A = AH . Now the form induced by b on A s ArAH is a nondegenerate alternating bilinear form, and we can proceed as before. Žc. W affords the G-character l ? 1 G with l e¨ en. Then taking G9 s '2 G*, one sees that G9 is a G-invariant integral lattice and G9*rG9 , Gr2G* affords the G-character l ? 1 G . Now apply the arguments of case Žb. to G9. v

Ž . EXAMPLE 2.6. Let G s Vq 8 2 be the commutator subgroup of the orthogonal group of dimension 8 with maximal Witt index. Let x be the unipotent character of G corresponding to the symbol Ž 0 1}2 3 .. Then x g IrrŽ G ., deg x s 28. Furthermore, the reduction x mod p is irreducible if p ) 2, and x mod 2 s 2 ? 1 G q b , where b g IBr 2 Ž G .. It can easily be shown Žsee, e.g., w20, Section 11.5x. that a QG-module with character x contains an odd unimodular lattice L Žof rank 28.. In fact, L is the exterior square of the root lattice of type E8 . ŽBy the way, w20, Chap. 11x contains a construction of an odd unimodular lattice G of dimension 28, without roots and having the Žfull. automorphism group Ž C2 . 8 ? ŽŽ C2 . 3 ? SL 3 Ž2.... Before giving an implicit proof of Theorem 1.1, we introduce some more notation. Let p ' 3 mod 4 be a prime and n an odd integer. Let c denote

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one of the Weil characters of S s Sp 2 nŽ p . of degree Ž p n q 1.r2. Inducing c from S to G s Sp 2 nŽ p . ? 2, one gets an irreducible Q-valued character x. LEMMA 2.7. Keep the abo¨ e notation. Then x is afforded by a QG-module Ž of dimension p n q 1.. Moreo¨ er, det V s Q 2 . v

Proof. Let q s p n. Then we can identify W s Fp2 n with Fq2 , and endow W with the symplectic form ² u, ¨ : s trŽ ad y bc ., where u s Ž a, b ., ¨ s Ž c, d ., a, b, c, d g Fq , and tr stands for the trace map tr: Fq ª Fp . Set H s GL2 Ž q ., and let the group R s  w g H < det w g Fp 4 , SL 2 Ž q . ? C py1 v

act on W in a natural way. Clearly, this action embeds R in the conformal ˆ s CSpŽW .. Let T denote the central subgroup symplectic group G  diagŽ l, l. < l g Fq 2 4 , CŽ qy1.r2 of H. Then the assumption n is odd implies that T l R s K, where K , CŽ py1.r2 is contained in the centre of ˆ and RT s H. Hence, x < R can be viewed as a faithful character of G, RrK s RrŽT l R . , HrT and so as a character, say r , of H Žwith kernel T .. The restriction of x to the subgroup R9 s SL 2 Ž q . is the sum of two irreducible Weil characters of degree Ž q q 1.r2 of R9. Inspecting the Ž . character table of H Žcf. w9x. we see that r s Ind H B m , where v

Bs

½ž

a 0

b d

/

a, d g Fq , c g Fq v

5

is a Borel subgroup of H, and the linear character m sends Ž 0a db . to d Ž a., d the quadratic character of Fq . In particular, r is rational Žthat is, it is afforded by a Q H-module, say V . and absolutely irreducible. ˆ , G. Thus we can view HrT as a subgroup of Next observe that GrK G and then r s x < H r T . The above discussion ensures that the restriction of x to the subgroup HrT is rational and absolutely irreducible. Besides, x itself is Q-valued. Now a standard lemma Žsee, for instance, w20, Lemma 8.3.1x. says that x is also rational, i.e., the QŽ HrT .-module structure of V can be extended to a QG-module structure. The explicit description of r shows that HrT stabilizes a unimodular lattice Žwith Gram matrix diagŽ1, 1, . . . , 1.. in V. Since both G and HrT act absolutely irreducibly on V, the invariant form is essentially unique, and we conclude that det V s Q 2. v

v

Proof of Theorem 1.1. Let c denote again one of the Weil characters of S s Sp 2 nŽ p . of degree Ž p n q 1.r2. By Lemma 2.7 we have a QG-module

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V with character x such that x < S s c q c , and a G-invariant integral lattice L in V. Recall that G s Sp 2 nŽ p . ? 2 is generated by S and some involution q , which induces an outer automorphism of S. The case p n s 3 is trivial: G , GL2 Ž3. stabilizes a root lattice of type D4 , and so we are done. Therefore we may suppose in what follows that p n ) 3. Then G has no normal subgroups of index 3 or 6. It is well known Žsee, e.g., w14, 15x. that c mod r g IBr r Ž S . for any odd prime r. Furthermore, c mod 2 s 1 S q h for some h g IBr 2 Ž S .. If x is a regular unipotent element of S, then c Ž x . s Ž1 " p ny 1 yp .r2 Žsee w14x.. Furthermore, q interchanges the S-conjugacy classes of x and xy1 . Therefore, x mod r g IBr r Ž G . for any prime r, r / 2, p. Furthermore, x mod 2 s 2 ? 1 G q m for some m g IBr 2 Ž G .. Finally, c mod p is of symplectic type Žsee w14, 15x.. These arguments show that the pair Ž x , G . satisfies the conditions Ži., Žii. of Proposition 2.5. Furthermore, by Lemma 2.7, det L is a square. Now we can apply Proposition 2.5 and get thereby a G-invariant unimodular lattice G. Standard arguments show that min G G 3 if n ) 1. If V contains an S-invariant even unimodular lattice D, then 8 divides dim D s dim V s p n q 1. Conversely, suppose p n ' y1 mod 8. It is well known and easy to see that V contains even unimodular lattices which are invariant under a subgroup of index 2 of G Žsee, e.g., Theorem 2.8 of w12x.. But the only subgroup of index 2 of G is S, so these lattices are S-invariant.

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So far, Theorem 1.1 has been proved in an implicit way. An explicit construction for G-invariant odd unimodular Žresp. S-invariant even unimodular. lattices will be given by Theorem 3.6 Žresp. Proposition 5.11.. Remark 2.8. In this paper, we focus our investigations on the 2-part of the discriminant group of integral lattices. Clearly, the approach exposed here can also be applied to other primes. For example, consider the unique irreducible complex character x of degree 26 of G s3 D4 Ž2.. Then QŽ x . s Q, indŽ x . s 1. Moreover, x mod p is absolutely irreducible for all primes p other than 3, and x mod 3 s 1 G q b for a certain b g IBr 3 Ž G .. From this it follows that x has Schur index 1 over Q. Now arguing as in the proofs of Lemma 2.1 and Proposition 2.4, we see that G stabilizes an Euclidean lattice L of rank 26 and determinant 3. Clearly, L is even and has no roots. In fact, one can show that min L s 3 and AutŽ L . s C2 = Ž G ? 3.. Moreover, Borcherds and Elkies have independently shown that L is the unique even lattice of rank 26 and determinant 3 with no roots. This lattice is investigated in detail in w11x.

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3. LAGRANGIANS, SYMPLECTIC SPREADS, AND GEOMETRIC STRUCTURE OF THE LATTICES Consider an arbitrary G-invariant Žintegral. lattice L in V. Consider also a natural Fp S-module W s Fp2 n endowed with a nondegenerate symplectic form ² ? , ? :. Fix some symplectic basis Ž e1 , . . . , e n , f 1 , . . . , f n . of W. 0 . One may suppose that q has matrix Ž E0n yE in this basis. A Lagrangian is n a maximal totally isotropic subspace in W. Following w1x, we consider them oriented, i.e., equipped with an equivalence class of bases. Two bases Ž l 1 , . . . , l n . and Ž lX1 , . . . , lXn . of a given Lagrangian are equivalent, i.e., define the same orientation, if there exists g g GLnŽ p . with det g g Fp 2 such that g Ž l i . s lXi , 1 F i F n. To each Lagrangian L we associate the following two subgroups: v

G Ž L . s  w g G < w Ž L . s L4 , PROPOSITION 3.1.

S Ž L . s  w g G Ž L . < det Ž w < L . g Fp 2 4 . v

For any Lagrangian L, the set

L Ž L . s  ¨ g L < ;w g S Ž L . , w Ž ¨ . s ¨ 4 is an one-dimensional Z-module. Proof. Ž1. Without loss of generality, one can take L s ² e1 , . . . , e n : F p with the basis Ž e1 , . . . , e n .. Denote P s St S Ž L . , C pnŽ nq1.r2 ? GLn Ž p . ,

Q s S Ž L . , R s P l Q.

A model for the Weil representation of S with character c is described in w15x. From this description it follows that c < P s d q z , where z is a P-character of degree Ž p n y 1.r2 and

dŽw. s

ž

det Ž w < L . p

/

.

Here w g P and Ž mrp. denotes the Legendre symbol. In particular, c < R s 1R q z < R. Ž2. We claim that z < R g IrrŽ R .. For, denote the largest normal p-subgroup Op Ž P . of P by E. Then one can identify E with the space of symmetric matrices of degree n over Fp . Furthermore, PrE , GLnŽ p . acts on E by the rule A( X s A ? X ? tA for A g GLnŽ p . s PrE, X g E. Obviously E ­ Ker z . So it is sufficient to show that every R-orbit on the set IrrŽ E . _  1 E 4 has length

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G Ž p n y1.r 2, or equivalently, every R-orbit on the set E*_  04 has length G Ž p n y1.r 2. Here E is viewed as a Fp-space, and E* stands for the dual space. Now we make the following two observations: }As p ' 3 mod 4 and n is odd, PrE is the direct product of RrE and a cyclic group of order 2 which acts trivially on E; }One can identify the GLnŽ p .-module E* with E itself, but endowed with the action A X s tAy1 ? X ? Ay1 , where A g GLnŽ p ., X g E. ŽIndeed, each element f g E* can be realized as the map f s f M : X ¬ TrŽ X ? M . for a uniquely determined M g E. Now we write down the action of A g GLnŽ p . on E*: v

Ž A f . Ž X . s f M Ž Ay1 ( X . s Tr Ž Ay1 ? X ? tAy1 ? M . v

sTr Ž X ?t Ay1 ? M ? Ay1 . s f A

v

M

Ž X . ..

Therefore our claim is equivalent to that each GLnŽ p .-orbit on the set E_  04 has length G Ž p n y 1.r2. Consider a GLnŽ p .-orbit O on E_  04 and X g O . Then the stabilizer of X in GLnŽ p . is nothing else but the isometry group of the symmetric bilinear from on Fpn with the matrix X. It is now not difficult to show that the cardinality of O is Ž p n y 1.r2 if rank X s 1, and strictly greater than Ž p n y 1.r2 if rank X ) 1. We remark that this claim also holds for n even. Ž3. Decompose V mQ C into a sum U [ U1 [ U2 of three R-submodules, with character 2 ? 1 R , z and z , respectively. Remark that R contains a regular unipotent element x and z Ž x . s Žy1 " p ny 1 yp .r2. Furthermore, Q s ² R, q :, and q normalizes R. Therefore q fixes U, and either leaves both U1 , U2 invariant or interchanges them. But q interchanges the S-conjugacy classes of x and xy1 , and z Ž xy1 . s z Ž x . / z Ž x .. This means q interchanges U1 and U2 . Besides that, x Žq . s 0. Hence, q acting on U has trace 0. We have shown that the fixed point subspace

'

F˜ s  ¨ g V mQ C < w Ž ¨ . s ¨ ;w g Q 4 s U l Ker Ž q y 1 . has dimension 1. From Lemma 3 of w7x it follows that the subspace F s  ¨ g V < w Ž ¨ . s ¨ ;w g Q4 has also dimension 1 Žover Q.. Since V s L mZ Q, we arrive at the conclusion that LŽ L. is an 1-dimensional Z-module. In view of Proposition 3.1, we denote by ¨ Ž L. a generating element of the Z-module LŽ L. for any Lagrangian L. Then ¨ Ž L. is determined uniquely up to sign. It is clear that LŽ L. is stabilized by GŽ L.. Namely,

w Ž ¨ Ž L. . s

ž

det Ž w < L . p

/

? ¨ Ž L.

Ž 1.

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for w g GŽ L.. If we consider Lagrangians oriented, then one can set

¨ ŽyL. s y¨ Ž L. for the opposite Lagrangian yL, corresponding to a

given oriented Lagrangian L. More precisely, we fix an oriented Lagrangian L0 with a basis Ž e1 , . . . , e n . and fix a generating vector ¨ 0 s ¨ Ž L0 . of LŽ L0 .. For an arbitrary oriented Lagrangian M with a basis Ž f 1 , . . . , f n ., we find an element n M g G such that n Ž e i . s f i for all i, and set ¨ Ž M . s n M Ž ¨ 0 .. This definition is independent of the choice of n M . For, if Ž . g g G and g Ž e i . s f i for all i, then ny1 M g e i s e i for all i. Therefore by Ž1. we obtain ny1 Ž . Ž . Ž . M g ¨ 0 s ¨ 0 , that is, g ¨ 0 s n M ¨ 0 . Moreover, for any Ž . h g G with h L0 s M, we have h Ž ¨ Ž L0 . . s

ž

< det Ž Ž ny1 M h. L0 . p

/

?¨Ž M..

y1 Ž . Ž . Ž . ŽIndeed, ny1 M h L 0 s L 0 , therefore n M h g G L 0 , and by 1 we have y1 Ž Ž n M h ¨ L0 .. s a ¨ Ž L0 ., hŽ ¨ Ž L0 .. s an M Ž ¨ Ž L0 .. s a ¨ Ž M ., where a s ŽdetŽŽ ny1 .< . . . M h L 0 rp .

PROPOSITION 3.2. If the intersection of two Lagrangians L and L9 has e¨ en dimension o¨ er Fp , then ¨ Ž L. and ¨ Ž L9. are orthogonal to each other. Proof. Again consider the symplectic basis Ž e1 , . . . , e n , f 1 , . . . , f n .. If dimŽ L l L9. s 2 k, k a nonnegative integer, then without loss of generality one can suppose that L s ² e1 , . . . , e n : F p ,

L9 s ² e1 , . . . , e2 k , f 2 kq1 , . . . , f n : F p .

Clearly that q is contained in both of GŽ L., GŽ L9.. Furthermore, detŽq < L . s 1, but detŽq < L9 . s y1. Due to Ž1. one then has q Ž ¨ Ž L.. s ¨ Ž L., q Ž ¨ Ž L9.. s y¨ Ž L9.. Therefore, Ž ¨ Ž L., ¨ Ž L9.. s Žq Ž ¨ Ž L.., q Ž ¨ Ž L9... s y Ž ¨ Ž L., ¨ Ž L9.., i.e. Ž ¨ Ž L., ¨ Ž L9.. s 0. LEMMA 3.3. Let L and M be arbitrary Lagrangians. Then <Ž ¨ Ž L., ¨ Ž M ..< depends only on the dimension of L l M Ž and on the choice of the norm Ž ¨ Ž L., ¨ Ž L.... In other words, there exists nonnegati¨ e constants a k , k s 0, 1, . . . , n, such that <Ž ¨ Ž L., ¨ Ž M ..< s a k whene¨ er dimŽ L l M . s k. Proof. Consider Lagrangians L9, M9 with dimŽ L l M . s dimŽ L9 l M9.. We have to show that Ž ¨ Ž L9., ¨ Ž M9.. s "Ž ¨ Ž L., ¨ Ž M ... It is clear that there exists an element w g S mapping L into L9 and M into M9. One readily verifies that w g wy1 g SŽ L9. whenever g g SŽ L.. Applying Ž1. we have g wy1 Ž ¨ Ž L9 . . s wy1 ? w g wy1 Ž ¨ Ž L9 . . s wy1 Ž ¨ Ž L9 . .

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for each g g SŽ L.. By Proposition 3.1 this implies that wy1 Ž ¨ Ž L9.. s "¨ Ž L., i.e., w Ž ¨ Ž L.. s "¨ Ž L9.. Similarly, w Ž ¨ Ž M .. s "¨ Ž M9.. In particular, Ž ¨ Ž L9., ¨ Ž M9.. s " Ž ¨ Ž L., ¨ Ž M ... In the notation of Lemma 3.3, Proposition 3.2 means that a k s 0 for all even values of k. Our next goal is to determine a k for odd k. Recall that a symplectic spread of W is a collection p s Wi < 1 F i F pn q 14 consisting of p n q 1 maximal totally isotropic subspaces such that p nq1 Wi s W. The so-called standard, or desarguesian, symplectic spread Di of W can be constructed in the following way. Identify W with Fq2 , q s p n, and endow W with the symplectic form ² u, ¨ : s trŽ ad y bg ., where u s Ž a , b ., ¨ s Žg , d ., a , b , g , d g Fq , and tr stands for the trace form tr: Fq ª Fp . Then

p D s  W l < l g Fq j  ` 4 4 , where W ` s Ž0, a . < a g Fq 4 , W l s Ž a , la . < a g Fq 4 for l g Fq , is the required spread. One may suppose that W 0 s ² e1 , . . . , e n : F p , W ` s ² f 1 , . . . , f n : F p . For a given symplectic spread p s Wi 4 , its automorphism group AutŽp . is defined as the group  w g G s Sp 2 nŽ p . ? 2 < ; i ' j s.t. w ŽWi . s Wj 4 . For example Žsee w20, Lemma 1.2.6x., Aut Ž p D . s SL 2 Ž q . ? Cn ? C2 , the extension of SL 2 Ž q . first by the Galois group of the extension FqrFp and then by the element q . Set L Ž p . s ² ¨ Ž L . < L g p :Z . Then, by Proposition 3.2, LŽp . is a sublattice of L of determinant n Ž a n . p q1 , where a n s Ž ¨ Ž L., ¨ Ž L.. as in Lemma 3.3. Now we consider the standard symplectic spread p D and project ¨ Ž M ., M a fixed Lagrangian, to the orthogonal basis Ž ¨ ŽW l ..: ¨Ž M. s

Ý l gF qj ` 4

zl¨ Ž W l . .

Ž ¨ Ž M ., ¨ ŽW l .. and so It is obvious that zl s ay1 n

Ý l gF qj ` 4

Ž¨ Ž M . , ¨ ŽW l. .

2

s a2n .

Ž 2.

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PROPOSITION 3.4. In the notation of Lemma 3.3 one has a k spyŽ nyk .r2 ? a n for any odd k, 1 F k F n. Proof. We shall proceed by induction on n s 1, 3, . . . . Ž1. First we take M s ² e1 , f 2 , . . . , f n : F p and write e1 s Ž e, 0. for e g Fq . Then M l W ` s ² f 2 , . . . , f n : has dimension n y 1. Furthermore, for an arbitrary l g Fq one has v

M l W l s  Ž xe, l xe . < x g Fp , ² Ž 0, l e . , e1 : s 0 4 s  Ž xe, l xe . < x g Fp , tr Ž l e 2 . s 0 4 . Therefore, dimŽ M l W l . is equal to 1 for just p ny 1 values of l g Fq , and 0 for the other l’s. Applying Ž2., one has p ny 1 a12 s a2n , i.e., a1 s pyŽ ny1.r2 a n . Thus we have proved Proposition 3.4 for the cases: Ža. k s 1, n G 1; Žb. n s 3, 1 F k F 3. In particular, the induction base n s 1, 3 has been established. Ž2. For the induction step we suppose n G 5. We already proved the desired relation of a1. Put W9 s ² e1 , . . . , e ny2 , f 1 , . . . , f ny2 : F p , W0 s ² e ny1 , e n , f ny1 , f n :F p , U s ² e ny1 , e n : F p , and introduce the following subgroups in S: B s St S ŽW9., S9 s  w g B < w < W 0 s 1W 0 4 , S0 s  w g B < w W 9 s 1W 94 , C s S0 l St S ŽU ., K s S9 = C. Then S9 , SpŽW9. s Sp 2 ny4 Ž p ., S0 , SpŽW0 . s Sp4Ž p ., C , Ž C p . 3 ? GL2 Ž p ., B s S9 = S0. We also set G9 s ² S9, q : , S9 ? 2, H s ² K, q : s G9 ? C , K ? 2. It is well known that c < B s c 9 m c 0 q t 9 m t 0, where c 9 Žresp. t 9. is an irreducible Weil character of S9 of degree Ž p ny 2 q 1.r2 Žresp. Ž p ny 2 y 1.r2.. Furthermore, c 0 Žresp. t 0 . is an irreducible Weil character of S0 of degree Ž p 2 q 1.r2 Žresp. Ž p 2 y 1.r2.. Arguing as in the proof of Proposition 3.1, we are convinced that a [ t 0 < C g IrrŽ C ., and c 0 < C s d q b , where b g IrrŽ C . and d Ž w . s ŽdetŽ w < U .rp . for w g C. ŽIn particular, d Ž1. s 1.. Thus

c
x < K s Ž c 9 q c 9. m d q Ž c 9 m b q c 9 m b . q Žt 9 m a q t X m a . .

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127

Observe that q acts on S9 as an outer automorphism, and q interchanges the characters c 9 and c 9. Furthermore, C e H. Consequently, x < H has a unique irreducible constituent in which C acts by scalars. This constituent X affords K-character Ž c 9 q c 9. m d and G9-character x 9 s c 9G . Ž3. Next we consider the Z-submodule L9 s ² ¨ Ž L . < L s L9 [ U, L9 a Lagrangian in W9: Z in L. ŽThe symplectic form on W9 is inherited from the one on W.. Clearly that H leaves L9 fixed. Moreover, let L s L9 [ U, L9 a Lagrangian in W9 and w g C. Then w Ž L. s L. Hence, due to Ž1. the subgroup C acts on L9 as scalars Žand the corresponding character is dim Z L9 ? d .. By the result of Ž2., dim Z L9 s x 9Ž1. s p ny 2 q 1. Furthermore, the lattice L9 acted on by G9 s Sp 2 ny4Ž p . ? 2 affords the Weil character x 9. If we denote G9Ž L9. s StG9Ž L9., then, of course, G9Ž L9. s GŽ L. l G9 for L s L9 [ U. In other words, W9, L9, L9 and ¨ 9Ž L9. Ža generating vector of L9Ž L9.; cf. Proposition 3.1. play the same roles for G9 as W, L, L, and ¨ Ž L. do for G. Observe that there is a nonzero rational scalar a such that ¨ 9Ž L9. s "a ¨ Ž L.. Indeed, ¨ Ž L. g L9 by the definition of L9, and ¨ Ž L. is obviously fixed by S9Ž L9.; hence ¨ Ž L. g L9Ž L9. and ¨ 9Ž L9. s "a ¨ Ž L. for some a g Q . Now we can apply the induction hypothesis to G9 and L9. In doing so we consider two arbitrary Lagrangians L9, M9 of W9 with dimŽ L9 l M9. s k. Then for L s L9 [ U, M s M9 [ U one has dimŽ L l M . s k q 2, which implies that aXk s <Ž ¨ 9Ž L9., ¨ 9Ž M9..< s a 2 <Ž ¨ Ž L., ¨ Ž M ..< s a 2 a kq 2 . By the induction hypothesis, for k odd we have v

a 2 a kq 2 s aXk s p Ž ny2yk .r2 aXny2 s a 2 p Ž nyŽ kq2..r2 a n , i.e., a kq 2 s p Ž nyŽ kq2..r2 a n . Thus we have proved the desired relation for a l with l s 3, 5, . . . , n. The induction step is over. COROLLARY 3.5. Then

Rescale the ¨ Ž L.’s such that Ž ¨ Ž L., ¨ Ž L.. s p Ž ny1.r2 .

Ž ¨ Ž L. , ¨ Ž M . . s

½

"p Ž ky1.r2 , 0,

dim Ž L l M . s k ' 1 mod 2 dim Ž L l M . ' 0 mod 2.

Now we in a position to exhibit explicitly a G-invariant odd unimodular lattice in V. THEOREM 3.6. For e¨ ery Lagrangian L, choose a ¨ ector ¨ Ž L. in V mQ R fixed by SŽ L. and such that Ž ¨ Ž L., ¨ Ž L.. s p Ž ny1.r2 . Then the lattice D s DŽ p, n. generated by all ¨ Ž L., D s ² ¨ Ž L . < L a Lagrangian in W : Z , is a G-in¨ ariant odd unimodular lattice.

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Proof. First we consider some G-invariant integral lattice L and choose

¨ 9Ž L. to be generating vector of the Z-module LŽ L.. Then, according to Propositions 3.2 and 3.4, Ž ¨ 9Ž L., ¨ 9Ž M .. s 0 if k s dimŽ L l M . is even and Ž ¨ 9Ž L., ¨ 9Ž M .. s p Ž ky1.r2 a1 if k is odd. Here a1 is some natural integer. Now we set ¨ Ž L. s a1y1 r2 ¨ 9Ž L. for all Lagrangians L. Clearly, ¨ Ž L. g V mQ R, Ž ¨ Ž L., ¨ Ž L.. s p Ž ny1.r2 and ¨ Ž L. is S Ž L.-stable. ŽWe could assume ¨ Ž L. g V by means of rescaling the scalar product on V by the scalar a1y1 .. Furthermore, Ž ¨ Ž L., ¨ Ž M .. g Z for any L, M. We see that

D, as defined in the theorem, is a G-invariant integral lattice. Moreover, if p D denotes the standard symplectic spread, then D contains the sublattice n G s DŽp D . s ² ¨ Ž L. < L g p D : Z of determinant p Ž ny1.Ž p q1 .r2 . In particular, det D is a power of p: det D s p m for some nonnegative integer m. If m s 0, we are done. Suppose that m G 1. Then consider the form Ž x, y . p s Ž x, y . mod p on DrpD Žsee the proof of Proposition 2.4.. As p divides det D, Ž?, ? . p is degenerate on DrpD. This means that pD is a proper sublattice of D l pD*. If D l pD* s D, then D : pD*; in particular, Ž ¨ Ž L., ¨ Ž M .. g pZ for all L, M, contrary to the equality Ž ¨ Ž L., ¨ Ž M .. s "1 for dimŽ L l M . s 1. Therefore, D > D l pD* > pD. One may then suppose that DrŽ D l pD*. affords the S-character c mod p. Since DrŽ D l pD*. supports the S-invariant nondegenerate symmetric bilinear form Ž?, ? . p , c mod p is of quadratic type, again a contradiction. Recall from Theorem 1.2 that our construction actually gives all G invariant odd unimodular lattices in V. Remark 3.7. The Gram matrix for the generating system,

 ¨ Ž L . < L a Lagrangian in W 4 , of the lattice DŽ p, n. has been written down in Corollary 3.5, but up to sign. The sign of Ž ¨ Ž L., ¨ Ž M .. for any pair Ž L, M . of oriented Lagrangians has been determined in w1x for the case p n s 27, and in w26x, for the case n s 3. 4. SYMPLECTIC SPREADS AND SELF-DUAL CODES: n s 3 In this section we restrict ourselves to the case n s 3. We maintain the notation G s Sp6 Ž p . ? 2. We continue the investigation of the G-invariant odd unimodular lattice D s DŽ p, 3. obtained in Theorem 3.6. The generating vectors ¨ Ž L. now have norm Ž ¨ Ž L., ¨ Ž L.. s p, and D contains a p-scaled unit lattice G, spanned by N [ p 3 q 1 pairwise orthogonal vectors of norm p Žfor instance, the ¨ Ž L., where L runs over a symplectic spread.. Therefore, D can be described Žnoncanonically. by a subspace

SYMPLECTIC GROUPS, SPREADS, CODES, AND LATTICES

129

C Žs DrG . ; G*rG s Ž1rp . GrG ( FpN , that is, by a linear code over Fp . The next proposition describes the situation more precisely. Two codes are called equi¨ alent if they can be mapped onto each other by a monomial matrix with entries "1. Notice that, for p ) 3, one has another natural notion of equivalence of codes defined by arbitrary monomial matrices with entries in Fp . This coarser equivalence relation however does not apply here. v

PROPOSITION 4.1. There exists an injecti¨ e mapping p ¬ C s C Žp . from the set S of all G-orbits of symplectic spreads p of W s Fp6 to the set C of all equi¨ alence classes of self-dual codes C of length p 3 q 1 o¨ er Fp with the property that AutŽ C . can be embedded in G s Sp6 Ž q . ? 2. Moreo¨ er, AutŽp . s AutŽ C Žp ... Proof. If p is a symplectic spread, set DŽp . [ ² ¨ Ž L. < L g p : Z : D, and let C Žp . : FpN s Ž c L .L g p < c L g Fp 4 correspond to DrDŽp . ; Ž1rp . DŽp .rDŽp . under the basis isomorphism FpN ª Ž1rp . DŽp .rDŽp .. The unimodularity of D means simply that C is a self-dual code over Fp . If, as usual, the elements of p are considered as ordinary, nonoriented Lagrangians, the ¨ Ž L.’s are only defined up to sign, but all choices lead to equivalent codes. Every enumeration of L, that is, every bijection n :  1, . . . , N 4 ª p , transform C Žp . into a subspace C Žp , n . ; FpN . Different choices of n permute the "¨ Ž L.. Thus the equivalence class of C Žp , n . is uniquely determined. If two spreads p and r give rise to equivalent codes, then one has a bijection p ª r and sign changes in the components such that the induced isomorphism DŽp . ª DŽ r . extends an automorphism F of D. Necessarily F g G, since G is the whole automorphism group of D Žsee Corollary 5.10 below.. It is clear that F Žp . s r , as desired. Similarly, AutŽp . s AutŽ C Žp ... COROLLARY 4.2. For e¨ ery odd prime p, the mapping mentioned in Proposition 4.1 gi¨ es at least two Ž nonisomorphic. self-dual codes of length p 3 q 1 o¨ er Fp . For the proof, one just has to recall the known fact that the symplectic space W s Fp6 has at least two nonisomorphic symplectic spreads. We have already exhibited the construction of the standard symplectic spread p D . This spread is desarguesian Žor regular .; that is, its kernel, Ker p s  w g End Ž W . < ; i , w Ž Wi . : Wi 4 , Žif p s Wi < 1 F i F p 3 q 14. is isomorphic to Fp 3 . It is well known that the kernel of any symplectic spread is a finite field of order F '< W < , and that any two desarguesian spreads are isomorphic; see, e.g., w20, Section 1.2x. A nondesarguesian spread, p BK L , has recently been constructed by Bader,

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Kantor, and Lunardon w4x. In fact, their construction gives nondesarguesian spreads of arbitrary odd dimension Žover the kernel.. Let us recall this construction. Take any finite field E of odd order and a nontrivial automorphism s such that E has odd degree over the fixed field E s of s . Let F be a subfield of E s , and set l s wE : Fx. Then the subspaces

 Ž 0, y . < y g E 4 y1

 Ž x, mx s

q ms x s . < x g E 4 Ž m g E.

of the F-space E 2 , F 2 l form a spread that we denote by p BK L . This spread is symplectic with respect to the symplectic form ŽŽ a, b ., Ž c, d .. ¬ tr E r F Ž ad y bc .. It is related to one of Albert’s twisted fields; see w4x. Its kernel is equal to E s . Furthermore, the automorphism group of p BKL Žover F. is equal to ŽŽŽ C p . al ? C p aly1 . ? Cl . ? C2 , where Ž C p . al is the additive group of the field E of order p al , C p aly1 is the multiplicative group of E, and Cl s GalŽErF.. In our case, one simply takes E s Fp 3 , s be a generating element of GalŽErFp ., and F s E s s Fp . EXAMPLE 4.3. Let p n s 27. We first recall from w21x the upper bound 3w Nr12x q 3 for the minimum weight of a self-dual ternary code of block length N. Codes attaining this bound are called extremal; they have a uniquely determined weight enumerator. In our case N s 28 we have d F 9. For a code coming from a lattice with minimal norm 3, like the lattice DŽ3, 3., the minimal weight necessarily equals 3 Žsince a codeword of weight 3k gives rise to a vector of norm k .. Therefore, all codes coming from symplectic spreads as above are extremal and are known to posses 2184 codewords of weight 9. It had been known for a while Žsee, e.g., w20, Section 1.2x. that W s F 36 has at least two symplectic spreads, namely p D and p H with AutŽp H . s SL 2 Ž13.. The latter spread is related to the Hering translation plane of order 27 w16x. ŽIt can be shown Žsee w20, Theorem 1.2.7x. that this spread and the standard spread are the only symplectic spreads which produce translation planes of order p n admitting 2-transitive collineation groups.. From Corollary 4.2 we know that there exists one further spread p BK L . We sum up these results in the following corollary. COROLLARY 4.4. The mapping mentioned in Proposition 4.1 gi¨ es us precisely three Ž nonisomorphic. extremal self-dual ternary codes of length 28; namely, CD , CH , and C BKL with automorphism groups Ž SL 2 Ž27. ? C3 . ? C2 , SL 2 Ž13. and ŽŽŽ C3 . 3 ? C26 . ? C3 . ? C2 . The first code, CD was first constructed by Ward w33x, and, independently, W. Scharlau and Chen. At the moment when the authors were

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131

finishing this work, the second code, CH , and the third code, C BKL , seemed to be new. However, Huffman w17x has classified all extremal self-dual ternary codes of length 28 which admit automorphisms of order 7 Žresp. 13.. Clearly, CD and CH have automorphisms of order 7 and 13, C BKL has automorphisms of order 13. Remark 4.5. Return to the Bacher]Venkov unimodular lattice D of dimension 28 Ž DŽ3, 3. in our notation., and let M denote the set of all 2240 minimal vectors of D. Then for u, ¨ g M , u / "¨ one has Ž u, ¨ . g  0, "14 . In other words, M forms a Ž0, 1r3.-system in the sense of Neumaier w23x. Such systems have been studied in that paper; their relation to some Buekenhout geometries has also been explained. The lattice DŽ3, 3. is an example of unimodular lattices G of rank 28 with minimum 3 which possess the following property: ŽBV.: nal.

G has a subset of 28 minimal vectors which are pairwise orthogo-

This property has been investigated in detail first by Bacher and Venkov. In particular, they found some invariant of G which enables one to recognize property ŽBV. w2x. There is an obvious link between such lattices and extremal self-dual ternary codes of length 28. Thus from the classification of unimodular lattices of rank 28 by Bacher and Venkov w2x one might hope to extract a classification of extremal self-dual ternary codes of length 28. PROPOSITION 4.6. The minimum of the lattice DŽ p, 3. satisfies the estimates pq1 F min D Ž p, 3 . F p. 2 Proof. Consider the standard symplectic spread p D , the corresponding sublattice G s DŽp D ., and the corresponding self-dual code CD s DrG. Then AutŽ CD . s AutŽp D . s Ž SL 2 Ž q . ? 3. ? 2, where q s p 3. In particular, AutŽ CD . contains the subgroup H s SL 2 Ž q . ? 2, consisting of matrices of determinant "1 and order 2 over Fq . Our aim is to show that CD is just the self-dual quadratic residue code constructed by Ward in w33x. As a consequence, we get that the minimum weight of CD is at least Ž p 2 q 5.r2. Since 1 min D G min min G, min CD , p

½

5

we obtain the desired estimate min D G Ž p q 1.r2. Recall the construction of quadratic residue codes exposed in w33x. Let n be the quadratic character on Fq given by n Ž x . s 1 if x is a nonzero

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square, y1 is x is a nonsquare, and 0 if x s 0. First we consider the C-space A with the basic Žw z x < z g Fq j  `4.. Furthermore, we consider the group GL2 Ž q . with the following standard generators: Ra s

ž

a 0

0 , 1

/

Sb s

ž

b 0 , Ts 1 1

1 0

/

ž

y1 , 0

/

a g Fq , b g Fq . v

One can define an action of GL2 Ž q . on A via the following formulas: R a : w`x ¬ n Ž a.w`x, w z x ¬ w az x, Sb : w`x ¬ w`x, w z x ¬ w z q b x,

Ž3.

T : w`x ¬ w0x ¬ n Žy1.w`x, w z x ¬ n Ž z .wy1rz x. This representation is in fact induced from the character

ž

a 0

b ¬ n Ž a. d

/

Ž 4.

of a parabolic subgroup of GL2 Ž q ., which is the stabilizer of w`x. It is not difficult to see that the kernel of this action contains the central subgroup Zs

½ž

a 0

0 a

/

5

a g Fq 2 , CŽ qy1.r2 v

of GL2 Ž q .. Now let B be the Fq-space with the basis Žw z x < z g Fq j  `4., and let GL2 Ž q . act on B by the formulas of Ž3.. Ward w33x has classified all GL2 Ž q .-codes in B. In particular, if n s 3 Žthe case we are considering. then there exists a unique GL2 Ž q .-invariant self-dual code over Fp in B, say, C W Žthe middle code, in the terminology of w33x.. Ward has shown that min C W G Ž p 2 q 5.r2. Now we return to our assumption that q ' 3 mod 4. Then GL2 Ž q . can be written as the direct product of Z and H: GL2 Ž q . s Z = H, therefore in fact the spaces A and B are acted on by H. Recall that H permutes q q 1 pairs of vectors "¨ ŽW l ., l g Fq j  `4 , transitively Žsee the construction of p D .. The action of St H Ž "¨ ŽW 0 .4. on ¨ ŽW 0 . is given by more general formula Ž1.. Comparing Ž1. and Ž4., we are immediately convinced that one can identify

w ` x § ¨ Ž W 0 . , w 0 x § ¨ Ž W ` . , w z x § ¨ Ž Wy1 r z . , z g Fq

v

and then A with G mZ C, B with Ž G*rG . mF p Fq . But CD is an H-invariant self-dual code over Fp ; therefore the codes CD and CW are isomorphic.

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133

Remark 4.7. From w33x and the proofs of Propositions 4.6 and 3.4 we extract the following estimate for the minimum of the middle code CW of Ward: Ž p 2 q 5.r2 F min CW F p 2 . 5. PROOF OF THEOREM 1.2 Let D s DŽ p, n. again denote the G-invariant odd unimodular lattice obtained by means of Theorem 3.6. In Section 1 we have exhibited three G-invariant lattices, namely, D, D0 , and D 1. It is clear that these three lattices cannot be similar to each other if p n ) 3 Žjust look at their determinants.. The aim of this section is to show that any integral lattice G of rank p n q 1 with a faithful irreducible action of G is similar to one of the three above-mentioned lattices. In a forthcoming paper we show that G has precisely one faithful irreducible Q-valued character of degree p n q 1. Therefore, the CG-modules G m C and D m C are equivalent. By the Deuring]Noether theorem, the QG-modules G m Q and D m Q s V also are equivalent. Hence, without loss of generality one may suppose that G is a G-invariant sublattice in D. The case p n s 3 is trivial: every G-invariant lattice in V is isometrically similar to D, D0 , or D 1 , which are the root lattices of type B4 , D4 , and D4U , respectively. ŽObserve that the dual lattice D4U is isometric to the root lattice D4 .. Therefore, throughout this section we shall suppose that p n ) 3. First we study the reduction Vr s DrrD, r a prime. LEMMA 5.1. Assume r is an odd prime. Then the Fr G-module Vr s Grr G Ž G a G-in¨ ariant lattice in V . is irreducible if r / p. If r s p, then Vr is reducible. Proof. Ž1. We have mentioned in Section 2 that x mod r g IBr r Ž G . if r / 2, p. Therefore, Vr is irreducible if r / 2, p. Consider the reduction x mod p. Recall that x < S s c q c . It is shown in w15x that c mod p s c mod p s h is obtained by restricting the irreducible algebraic representation of Sp 2 nŽ Fp . with highest weight ŽŽ p y 1.r2. v n to S s Sp 2 nŽ p .. Furthermore, due to Lemma 2.6 from w30x, h is invariant under the action of the distinguished involution q . Therefore, G has just two irreducible Brauer characters h1 , h 2 with hi < S s h. In this case, x mod p s h1 q h 2 . Our claim will follow, if we show that the Frobenius endomorphism Ž p. stabilizes the hi ’s; that is, hiŽ p. s hi , i s 1, 2. In order to do it, we need to consider another double extension H , S ? 2, which is isoclinic to G. Ž2. We can embed S in T s Sp 2 nŽ p 2 . in the following way. In a ˜ of T consider a symplectic basis, natural 2 n-dimensional Fp 2-module W

Ž e1 , . . . , e n , f 1 , . . . , f n . ,

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that is, a basis in which the symplectic form is given as ² e i , e j : s 0, ² f i , f j : s 0, ² e i , f j : s d i, j . In this basis we can set W s ² e1 , . . . , e n , f 1 , . . . , f n : F p ,

Js

ž

« En

0 y1

«

0

En

/

.

Here « g Fp 2 is chosen with order 2Ž p y 1. and Em denotes the identity matrix of order m. Now we set S s T l EndŽU . , Sp 2 nŽ p ., H s ² S, J :. Then J 2 g S and J normalizes S; therefore H , Sp 2 nŽ p . ? 2. Suppose that g g CH Ž S . _S. Since the action of S on W is absolutely irreducible, g s l E2 n for some l g Fp 2 . Now from the inclusion g g H_S it follows that l2 s 1 and l« g Fp , which is impossible because « f Fp . We have shown that ZŽ H . s ZŽ S . s CH Ž S . s C2 . In other words, H is a double extension of S, which is isoclinic to G. Now passing the embedding S ¨ Sp 2 nŽ Fp . through T s Sp 2 nŽ p 2 ., one sees that h is extended to two absolutely irreducible Brauer characters m 1 , m 2 of H. We calculate the value of m 1 , m 2 at the element J. If one denotes e s expŽp irŽ p y 1.., then m 1Ž J . s Ý u g Iyn e < u< . Here

½

< < Iy n s u s Ž u1 , . . . , u n . u j g Z, u j F Ž p y 1 . r2,

n

Ý

5

u j ' 1 Ž mod 2 . ,

js1

Ž Ž . . and < u < s Ý j u j for u g Iy n . Recall that n p y 1 r2 is odd in our case. Set

½

< < Iq n s u s Ž u1 , . . . , u n . u j g Z, u j F Ž p y 1 . r2,

n

Ý

Sq n s

Ý

ugI q n

e < u< ,

Sy n s

Ý

ugI y n

e < u< .

Denote also u s cotŽpr2Ž p y 1... Then we have S1q s

u q uy1 2

,

S1y s

q q y y Sq nq 1 s S n S 1 q S n S 1 ,

u y uy1 2

,

q y y q Sy nq1 s S n S 1 q S n S 1 .

From this it follows that Sq n s

u n q uyn 2

,

In particular, m 1Ž J . s Ž u n y uyn .r2.

Sy n s

u n y uyn 2

.

5

u j ' 0 Ž mod 2 . ,

js1

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Assume p G 7. Then u ) 1; hence m 1Ž J . is a positive real number. Moreover, the Frobenius endomorphism e ¬ e p sends e to ye , 'y1 to y 'y1 , m 1Ž J . to ym 1Ž J . s m 2 Ž J .. We have shown that mŽi p. s m 3y i s m 3yi for i s 1, 2. Consider the case p s 3. Then u s 1, and so m 1Ž J . s m 2 Ž J . s 0. Since p n ) 3, n is at least 3, and we can find an element s of order 8 in S, which is conjugate in T to diag Ž z , z 3 , 1, . . . , 1, zy1 , zy3 , 1, . . . , 1 . in the basis Ž e1 , . . . , f n .. Here we choose a generating element z of F 9 such that « s z 2 . Setting j s expŽp ir4., one has e s j 2 . Then for the element J9 s Js we have v

m 1 Ž J9 . s

Ý

ugI y n

j 3 u1q5 u 2 e u 3q? ? ?qu n .

Partition the terms of this sum with u 3 / 0 into pairs, each of which contains two terms with the same u i ’s, i / 3. Because e q ey1 s 0, each of these pairs gives zero contribution into m 1Ž J9.. The same arguments are applicable to any index i G 3. Hence,

m 1 Ž J9 . s

j 3 u1q5u 2 s y2'2 .

Ý u 1 , u 2s0, "1; u 1qu 2'1 Žmod 2 .

This computation allows us to conclude that

m1 s m1 ,

m2 s m2 ,

mŽ1p. s m 2

if p s 3. Ž3. Next we consider a representation F: H ª GLŽ p nq1.r2 Ž Fp . with ˜ s  F Ž g ., g F Ž h. < g g S, Brauer character m 1. Put g s « Ž py1.r2 , and set G h g H_S4 . Because the representation F is faithful, and the central involution of S is represented in F by the multiplication to g 2 s y1, we ˜ is a double extension of S. Claim that G˜ , G. are convinced that G ˜ is generated by the subgroup  F Ž g . < g g S4 , S and For, observe that G any element g F Ž Jh., h g S. Since p ' y1 Žmod 4., a s « Ž pq1.r2 belongs y1 to Fp , and so the element t with matrix Ž a 0 En a 0En . in the basis Ž e1 , . . . , f n . belongs to S. Recall that G is generated by S and the involution q with 0 . matrix Ž E0n yE in the basis Ž e1 , . . . , f n .. One can verify that Ž Jt . 2 s y1 is n the central involution of S. Therefore, Žg F Ž Jt .. 2 s g 2 F Žy1. s 1 s q 2 . Furthermore, the actions of g F Ž Jt . and of q on S via conjugation are the ˜ , G. same. This means: G

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˜ gives us a representation C: G ª The isomorphism G , G GLŽ p nq1.r2 Ž Fp .. One may suppose that this representation affords Brauer character h1. Assume p G 7. Let h be the element in G with C Ž h. s g F Ž J .. Then h1Ž h. s 'y1 m 1Ž J .. The computations in item Ž2. show that h1Ž h. is purely imaginary, and that the Frobenius endomorphism leaves h1Ž h. fixed. Consequently, h1 s h1Ž p. s h 2 , h 2 s h 2Ž p. s h1 , h1 / h 2 . Finally, suppose p s 3. Let h denote the element in G with C Ž h. s g F Ž J9.. Then h1Ž h. s 'y1 m 1Ž J9. s y2'y2 . Therefore, h1Ž h. is purely imaginary, and the Frobenius endomorphism fixes h1Ž h.. Consequently,

h1 s h1Ž p. s h 2 , h 2 s h 2Ž p. s h1 , h1 / h 2 . In particular, Vp is reducible. Warning. In general, the Fr G-module Grr G depends on the choice of the lattice G. In order to get more information about the submodules of Vp , we need the following statement. PROPOSITION 5.2. The QG-module V contains a G-in¨ ariant lattice = with the following property: There exists a Z-linear endomorphism f : = ª = such that

f 2 Ž ¨ . s yp¨ , Ž f Ž u . , ¨ . s y Ž u, f Ž ¨ . . , Ž f Ž u . , f Ž ¨ . . s p Ž u, ¨ . for any u, ¨ g =. Furthermore, f commutes with the action of S s Sp 2 nŽ p .. Moreo¨ er, if p s 3 then there exists a G-in¨ ariant e¨ en lattice = such that the endomorphism v s Žy1 q f .r2 is in fact an automorphism of =, which centralizes the subgroup S s Sp 2 nŽ3. in AutŽ = .. Proof. Ž1. The equality x < S s c q c implies that the commuting algebra K s End S Ž V . s  w g End Q Ž V . < ;s g S, w ? s s s ? w 4 is isomorphic to the field QŽ c . s QŽ yp .. In particular, K contains an endomorphism f with f 2 s yp. Fix a G-invariant lattice G in V. Then it is clear that f Ž G . : a G for some rational number a . By changing the scalar product, we can convert = s G q f Ž G . into an integral lattice. Ž2. By its definition, f commutes with the action of S, therefore S stabilizes =. Furthermore,

'

f Ž = . s f Ž G . q f 2 Ž G . s f Ž G . q pG : =;

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i.e., f stabilizes =. Claim that = is also G-invariant. For, consider the distinguished involution q g G_S. For any s g S and l g K, qlqy1 g End Q Ž V ., and s ? qlqy1 s q ? qy1 sq ? l ? qy1 s q ? l ? qy1 sq ? qy1 s qlqy1 ? s. This means: qlqy1 g K. Now it is clear that the map Q: l ¬ qlqy1 is an automorphism of K, which leaves the subfield Q pointwise fixed. In other words, Q g GalŽKrQ.. If QŽ l. s l, then l centralizes S and q , which implies that l g End G Ž V . s Q. Therefore, the automorphism Q of the quadratic field K is not identity, i.e., qlqy1 s l for any l g K. In particular, fq s yqf . Hence, q Ž f Ž G .. s yf Žq Ž G .. s f Ž G ., and so q Ž = . s =. Ž3. Consider a new scalar product Ž u, ¨ .9 s Ž f Ž u., f Ž ¨ .., u, ¨ g =. For g g G one has f g s b g f with b s "1. Hence,

Ž gu, g¨ . 9 s Ž f g Ž u . , f g Ž ¨ . . s Ž b g f Ž u . , b g f Ž ¨ . . s Ž g f Ž u . , g f Ž ¨ . . s Ž f Ž u . , f Ž ¨ . . s Ž u, ¨ . 9, i.e., Ž?, ? .9 is G-invariant. But G acts irreducibly on V, therefore Ž f Ž u., f Ž ¨ .. s Ž u, ¨ .9 s g Ž u, ¨ . for some positive rational number g . Now p 2 Ž u, ¨ . s Ž ypu, yp¨ . s Ž f 2 Ž u . , f 2 Ž ¨ . . s g Ž f Ž u . , f Ž ¨ . . s g 2 Ž u, ¨ . , so in fact g s p. We have shown that Ž f Ž u., f Ž ¨ .. s pŽ u, ¨ . for any u, ¨ g =. From this identity we also get the skew-symmetry of f ; namely, 1

Ž f Ž u . , ¨ . s y Ž f Ž u . , f 2 Ž ¨ . . s y Ž u, f Ž ¨ . . . p

In particular, Ž f Ž u., u. s 0. As a consequence, we obtain that the dual lattice =* is f-invariant. For, if u g =*, then

Ž f Ž u . , = . s y Ž u, f Ž = . . : Ž u, = . : Z, i.e., f Ž u. g =*. Ž4. Finally, assume p s 3. Setting v s Žy1 q f .r2, we see that v 2 s vy1 s v s y1 y v . Since s v s v s for any s g S, and qv s vq , G leaves the lattice = s G q v Ž G . fixed. It is clear that v and vy1 also stabilize =. Furthermore, 4 Ž v Ž u . , v Ž ¨ . . s Ž yu q f Ž u . , y¨ q f Ž ¨ . . s Ž u, ¨ . y Ž f Ž u . , ¨ . y Ž u, f Ž ¨ . . q Ž f Ž u . , f Ž ¨ . . s 4 Ž u, ¨ .

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for any u, ¨ g =. In other words, v g AutŽ = .. Remark that Žany. v-invariant integral lattice = is even. Indeed, for any u g = one has Z 2 Ž u, v Ž u . . s u,

ž

yu q f Ž u . 2

/

sy

Ž u, u . 2

.

It is clear that the endomorphism f mentioned in Proposition 5.2 is uniquely determined in K up to sign. Therefore, in what follows one can speak about f-stable G-invariant lattices in V. Obviously, the lattices G and f Ž G . are isometrically similar. A crucial role in further arguments is played by the fact that our odd unimodular lattice DŽ p, n. is f-stable! The next several assertions are to prove this claim. Recall that, to any Lagrangian L we have associated two subgroups SŽ L. and GŽ L.. Set also RŽ L. s SŽ L. l Sp 2 nŽ p .. It is clear that every g g GŽ L. acts on L and on WrL. Consider the following linear character j L of GŽ L.:

j LŽ g . s

ž

det Ž g < W r L . p

/

for g g GŽ L.. Let L be an arbitrary G-invariant integral lattice lying in V. LEMMA 5.3.

For any Lagrangian L, the set

Ly Ž L . s  ¨ g L < ; g g G Ž L . , g Ž ¨ . s j L Ž g . ¨ 4 is an one-dimensional Z-module. Proof. Remark that j L < RŽ L. s 1 RŽ L. . In the proof of Proposition 3.1 we have shown that Ž x < RŽ L. , 1 RŽ L. .RŽ L. s 2. There we have also singled out some subspace U of dimension 2 of V mQ C, which affords the RŽ L.-character 2 ? 1 RŽ L. . The involution q acts on U with trace 0, and j LŽq . s y1. This means that U affords the S Ž L.-character 1 SŽ L. q j L < SŽ L. . Hence, Ž x < SŽ L. , j L < SŽ L. .SŽ L. s 1. From Lemma 3 of w7x it follows that the subspace F s  ¨ g V < ; g g SŽ L., g Ž ¨ . s j LŽ g . ¨ 4 has dimension 1 Žover Q.. But GŽ L. is generated by SŽ L. and the central involution of Sp 2 nŽ p ., so in fact F is a GŽ L.-module, with character j L . Since V s L mZ Q, we arrive at the conclusion that LyŽ L. is an 1-dimensional Z-module. In view of Proposition 3.1 and Lemma 5.3, we denote by ¨ Ž L. Žresp. uŽ L.. a generating element of the Z-module LŽ L. Žresp. LyŽ L.. for a given Lagrangian L. By Lemma 3.3, a k s <Ž ¨ Ž L., ¨ Ž M ..< depends only on k s dimŽ L l M ..

SYMPLECTIC GROUPS, SPREADS, CODES, AND LATTICES

139

LEMMA 5.4. Ži. Let L and M be arbitrary Lagrangians. Then the parameters bk s <Ž uŽ L., uŽ M ..< Ž resp. c k s <Ž uŽ L., ¨ Ž M ..<. depends only the dimension k of the intersection L l M. Žii. c k s 0 if k is odd. Proof. Ži. Consider Lagrangians L9, M9 with dimŽ L l M . s dimŽ L9 l M9.. We have to show that Ž uŽ L9., uŽ M9.. s "Ž uŽ L., uŽ M .. and Ž uŽ L9., ¨ Ž M9.. s "Ž uŽ L., ¨ Ž M ... It is clear that there exists an element w g S mapping L into L9 and M into M9. One readily verifies that w GŽ L. wy1 s GŽ L9., w SŽ L. wy1 s SŽ L9., and w RŽ L. wy1 s RŽ L9.. We already know that w : ¨ Ž L. ¬ "¨ Ž L9., ¨ Ž M . ¬ "¨ Ž M9.. Suppose g g GŽ L.. Then j LŽ g . s j L9Ž w g wy1 .. Furthermore, g wy1 Ž u Ž L9 . . s wy1 ? w g wy1 Ž u Ž L9 . . s wy1 Ž j L9 Ž w g wy1 . u Ž L9 . . s j LŽ g . wy1 Ž u Ž L9 . . . By Lemma 5.3 this implies that wy1 Ž uŽ L9.. s "uŽ L., i.e., w Ž uŽ L.. s "uŽ L9.. Similarly, w Ž uŽ M .. s "uŽ M9.. Hence, Ž uŽ L9., uŽ M9.. s "Ž uŽ L., uŽ M .., Ž uŽ L9., ¨ Ž M9.. s "Ž uŽ L., ¨ Ž M ... Žii. Again consider the symplectic basis Ž e1 , . . . , e n , f 1 , . . . , f n . of W. If dimŽ L l M . s k, k an odd integer, then without loss of generality one can suppose that L s ² e1 , . . . , e n : F p ,

M s ² e1 , . . . , e k , f kq1 , . . . , f n : F p .

Clearly q is contained in both of GŽ L., GŽ L9.. Furthermore, detŽq < W r L . sy1 and detŽq < M . s1. Therefore, q Ž uŽ L.. syuŽ L., q Ž ¨ Ž M .. s¨ Ž M .. Now we get

Ž uŽ L. , ¨ Ž M . . s Ž q Ž uŽ L. . , q Ž ¨ Ž M . . . s yŽ uŽ L. , ¨ Ž M . . , i.e., Ž uŽ L., ¨ Ž M .. s 0. Now we consider the standard symplectic spread p D , and project uŽ M ., M a fixed Lagrangian, to the orthogonal basis Ž ¨ ŽW l ..: uŽ M . s

Ý l gF qj ` 4

zl¨ Ž W l . .

Ž uŽ M ., ¨ ŽW l .. and so It is obvious that zl s ay1 n

Ý l gF qj ` 4

Ž uŽ M . , ¨ Ž W l . .

2

s a n bn .

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In particular, take M s W ` . Then dimŽ M l W l . is equal to n if l s `, and 0 if l g Fq . Therefore, 2

a n bn s p n Ž c 0 . .

Ž 5.

Next we apply Lemmas 5.3, 5.4 to the lattice L s = s D q f Ž D .. ŽWith respect to the scalar product defined on D, = can eventually be nonintegral; but it does not affect further arguments.. Observe that =Ž L. s DŽ L.. ŽIndeed, = = D implies ¨ Ž L. s k ¨˜Ž L., where k g Z and =Ž L. s ² ¨˜Ž L.: Z . In this case ¨ Ž L. g k =. Since D is generated by the ¨ Ž L.’s, D : k =. From this it follows that f Ž D . : f Ž k = . : k =. Thus = : k =, and so k s "1.. Still =yŽ L. can eventually differ from DŽ L.. In the following statement we denote by u ˜Ž L. a generating vector of =yŽ L.. LEMMA 5.5. For the lattice = s D q f Ž D . we ha¨ e u ˜Ž L. s "f Ž ¨ Ž L... In particular, the parameters a k , bk , c k of the lattice = satisfy the following relations: Ži. bk s pak for any k; Žii. c 0 s a1 s pyŽ ny1.r2 a n . Proof. Recall that g f s detŽ g < W . f g for any g g G. If g g GŽ L., then detŽ g < W . s detŽ g < L . detŽ g < W r L ., and so detŽ g < W . s ŽdetŽ g < L .rp . ? j LŽ g .. Therefore, by Ž1. one has g f Ž ¨ Ž L . . s det Ž g < W . f g Ž ¨ Ž L . . s det Ž g < W . ?

ž

det Ž g < L . p

/

f Ž ¨ Ž L . . s j LŽ g . f Ž ¨ Ž L . . .

This means: f Ž ¨ Ž L.. g =yŽ L.; hence f Ž ¨ Ž L.. s k ? u ˜Ž L. for some k g Z. Similarly, gf Ž u ˜Ž L . . s det Ž g < W . f g Ž u˜Ž L . . s det Ž g < W . ? j LŽ g . f Ž u ˜Ž L . . s

ž

det Ž g < L . p

/

fŽ u ˜Ž L . . ,

which implies that f Ž u ˜Ž L.. g =Ž L.. From this it follows that f Ž u˜Ž L.. s Ž . l ? ¨ L for some l g Z. In this case we have yp ? ¨ Ž L . s f 2 Ž ¨ Ž L . . s f Ž k ? u ˜Ž L . . s k f Ž u˜Ž L . . s kl ? ¨ Ž L . , i.e., kl s yp. Assume k / "1. Then k s "p, l s .1. In that case,

¨ Ž L. s .f Ž u ˜Ž L.. belongs to f Ž = .. But = is generated by the vectors ¨ Ž L. and the sublattice f Ž D . which is contained in f Ž = .. Therefore,

SYMPLECTIC GROUPS, SPREADS, CODES, AND LATTICES

141

we have seen that = : f Ž = .. Applying f once more again, we get = : f 2 Ž = . s p=, a contradiction. Hence k s "1, i.e., u ˜Ž L. s "f Ž ¨ Ž L... Next we take L, M such that dimŽ L l M . s k. Then bk s <Ž u ˜Ž L., u˜Ž M ..< s <Ž f Ž ¨ Ž L.., f Ž ¨ Ž M ...< s p <Ž ¨ Ž L., ¨ Ž M ..< s pak . Furthermore, by Ž5. one has 2

2

2

p n Ž c 0 . s a n bn s p Ž a n . s p n Ž a1 . , i.e., c 0 s a1. Recall Žcf. Propositions 3.2 and 3.4. that = satisfies a k s p Ž kyn.r2 a n for k odd, and a k s 0 for k even. By Lemma 5.5, bk s p Ž ky2yn.r2 a n for k odd, and bk s 0 for k even. Now we determine c k for k even. LEMMA 5.6. for k e¨ en.

For the lattice = s D q f Ž D . one has: c k s p Ž kq1yn.r2 a n

Proof. We shall proceed by induction on n s 1, 3 . . . . By Lemma 5.5, c 0 s a1 s p Ž1yn.r2 a n . Thus we have proved Lemma 5.6 for the case where k s 0 and n G 1. In particular, the induction base n s 1 has been established. For the induction step we suppose n G 3. We shall follows the proof of Proposition 3.4 and maintain the notation of that proof. There we have proved that x < H contains a unique irreducible constituent g , in which C acts as scalars; namely, the one whose restriction to K is equal to Ž c 9 q c 9. m d . Denote D9 s ² ¨ Ž L . < L s L9 [ U, L9 a Lagrangian in W9: Z , V9 s D9 mZ Q. We have also shown that V9 affords the H-character g . Now recall that the endomorphism f centralizes S. In particular, f centralizes K. Hence, the subspace f Ž V 9. affords the same K-character as of V9. Since q Ž V9. s V9 and qf s yfq , f Ž V9. is q-stable, that is, f Ž V 9. is an H-module. By the above observation, f Ž V9. also affords the H-character g . Thus V 9 and f Ž V9. are simple submodules of the Q H-module V, which afford the same character g . As g enters x < H with multiplicity 1, f Ž V9. s V9. We have seen that W9, V 9, D9, f and ¨ 9Ž L9. s "a ¨ Ž L. Ž a a positive rational number; see the proof of Proposition 3.4. play the same roles for G9 as W, V, D, f , and ¨ Ž L. do for G. ŽOf course, the scalar product on V9 is inherited from the one on V.. Therefore, we may apply the induction hypothesis to G9 and =9 s D9 q f Ž D9.. In doing so we consider two Lagrangians L9, M9 of W9 with dimŽ L9 l M9. s k is even. If uŽ L. s f Ž ¨ Ž L.., then by Lemma 5.5 u9Ž L9. s "a uŽ L. generates =9yŽ L9.. For L s L9 [ U, M s M9 [ U one has dimŽ L l M . s k q 2, which implies that aXk s a 2 a kq2 , cXk s <Ž u9Ž L9., ¨ 9Ž M9..< s a 2 <Ž uŽ L., ¨ Ž M ..< s a 2 c kq2 .

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SCHARLAU AND TIEP

By the induction hypothesis,

a 2 c kq 2 s cXk s p ŽŽ kq1.yŽ ny2..r 2 aXny2 s a 2 p Ž kq3yn.r2 a n , and so c kq 2 s p Ž kq3yn.r2 a n . Thus we have proved the desired relation for c l with l s 2, 4, . . . , n y 1. The relation c 0 s p Ž1yn.r2 a n was above established. The induction step is over. PROPOSITION 5.7. The odd unimodular lattice D s DŽ p, n. obtained by means of Theorem 3.6 is f-stable. Proof. Let u ˜Ž L. denote a generating vector of =yŽ L., where = s D q Ž . f D is considered w.r.t. the scalar product defined on DŽ L. by means of Theorem 3.6. Then Ž ¨ Ž L., ¨ Ž M .., Ž u ˜Ž L., u˜Ž M .., and Ž u˜Ž L., ¨ Ž M .. are all integers for any L, M. Namely, by Proposition 3.4 and Lemmas 5.4, 5.5, 5.6 we have a k s p Ž ky1.r2 , bk s p Ž kq1.r2 , c k s 0 for k odd, and a k s 0 s bk , c k s p k r2 for k even. In particular, u ˜Ž L. g D* for any Lagrangian L. By Ž . Theorem 3.6, D* s D; therefore u ˜ L g D. Recall that = is generated by D and the vectors u ˜Ž L.’s; hence we arrive at the conclusion that = s D. In other words, D is f-stable. Now we are in a position to give the following supplement to Lemma 5.1. PROPOSITION 5.8. The module Vp s DrpD has a unique nonzero proper G-submodule, and this submodule coincides with f Ž D .rpD. Moreo¨ er, if G is any G-in¨ ariant sublattice of D with the index Ž D : G . being a power of p, then there exists an integer k G 0 such that G s f k Ž D .. Proof. Ž1. Let A be any nonzero proper submodule in Vp . In the proof of Lemma 5.1 we have shown that the Brauer character afforded by A is hi for some i s 1, 2, and hi is not self-dual. In particular, A is absolutely irreducible, and A cannot support any G-invariant nondegenerate bilinear form. This forces A be totally singular with respect to Ž?, ? . p , the reduction modulo p of the scalar product. An example of such a submodule A is f Ž D .rpD. If n s 1, then due to w33x, A is a unique nonzero submodule of Vp Žand A is called the modular quadratic residue code.. Therefore from now on we suppose that n G 3. Assume that A is not unique. The Vp s A [ B for some submodules A, B. One may suppose that A, B afford Brauer character h1 , h 2 s h1 , respectively. Recall once more that A and B are totally singular with respect to Ž?, ? . p . Ž2. Let L be any Lagrangian. By Proposition 3.1, the subgroup SŽ L. fixes the vector ¨ Ž L.. Observe that ¨ Ž L. f pD; hence one can view ¨ Ž L. as a nonzero vector in Vp . Set W Ž L . s  ¨ g Vp < ;w g S Ž L . , w Ž ¨ . s ¨ 4 .

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For brevity, we denote by xSŽ L. the restriction of x mod p s h1 q h 2 to SŽ L., by x < RŽ L. the restriction of x mod p to RŽ L., by a the trivial character of SŽ L., by b the nontrivial character of degree 1 of S Ž L. with Ker b s RŽ L.. ŽThen b s j L < SŽ L. .. Since W Ž L. 2 ¨ Ž L., dim W Ž L. G 1. Claim that x < SŽ L. contains a with multiplicity two. For, consider the standard embedding SL nŽ p . ¨ RŽ L. : Sp 2 nŽ p .. Zalesskii’s reduction formula for Weil characters w35x enables one to set that 1 < S L n Ž p. enters Ž c mod p .< S L n Ž p. with multiplicity at most two. From this it follows that a < RŽ L. s 1 RŽ L. enters x < RŽ L. with multiplicity at most four. In the proof of Proposition 4.1 we have singled out some subspace U of V, which is acted on by S Ž L. with character a q b . Hence, if k denotes the multiplicity, with which a enters x < SŽ L. , then k F 4 y 1 s 3. But x mod p s h1 q h1 , therefore k must be even. In our case, 1 F k F 3. Consequently, k s 2. In particular, dim W Ž L. F 2. Ž3. Write ¨ Ž L. s a q b for a g A, b g B. Remark that a, b / 0. ŽAssume the contrary: a s 0. Then ¨ Ž L. g B for any Lagrangian L. As B is totally singular, Ž ¨ Ž L., ¨ Ž M .. g pZ for any L, M. This last condition contradicts Proposition 3.4.. Now SŽ L. fixes each of the subspaces A, B, therefore in fact a, b g W Ž L., and W Ž L. s ² a, b : F p has dimension 2. As n G 3, we have 0 s Ž ¨ Ž L . , ¨ Ž L . . p s Ž a q b, a q b . p s Ž a, a . p q Ž b, b . p q 2 Ž a, b . p s 2 Ž a, b . p , which implies that Ž a, b . p s 0. We have just shown that W Ž L. is totally singular with respect to Ž?, ? . p : W Ž L. : W Ž L. H . Besides that, the SŽ L.modules VprW Ž L. H and W Ž L.* are isomorphic. From this it follows that VprW Ž L. H affords the SŽ L.-character 2 a . But in that case, k G 4, a contradiction. Ž4. We have shown that f Ž D .rpD is the unique nonzero proper submodule of Vp . Now suppose that G is a G-invariant lattice with D = G = p aD for some nonnegative integer a. Then D = G = f b Ž D ., where b s 2 a. Choose minimal c G 0 with property G = f c Ž D .. We prove by induction on c that G s f k Ž D . for some k. This claim is obvious if c s 0. Assume c G 1. If G : f Ž D ., then D = fy1 Ž G . = f cy 1 Ž D .. By the induction hypothesis, fy1 Ž G . s f k Ž D ., G s f kq 1 Ž D ., and we are done. Suppose G ­ f Ž D .. In this case, Ž G q f Ž D ..rf Ž D . is a nonzero submodule of the irreducible G-module Drf Ž D .; therefore G q f Ž D . s D. Furthermore, Ž G q f 2 Ž D ..rpD is a nonzero submodule in Vp , hence G q f 2 Ž D . = f Ž D .. Thus we have D s G q f Ž D. : G q G q f2 Ž D. s G q f2 Ž D. ,

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i.e., G q f 2 Ž D . s D. Now, if c is even, then c G 2, and f cy 2 Ž G . s p Ž cy2.r2 G : G. Therefore,

f cy 2 Ž D . s f cy 2 Ž G q f 2 Ž D . . s f cy 2 Ž G . q f c Ž D . : G, contradicting the choice of c. If c is odd, then c G 1, and f cy 1 Ž G . s p Ž cy1.r2 G : G. Therefore,

f cy 1 Ž D . s f cy 1 Ž G q f 2 Ž D . . s f cy 1 Ž G . q f cq 1 Ž D . : G, again contrary to the choice of c. The induction step is over. LEMMA 5.9. The F 2 G-module V2 has precisely two nontri¨ ial proper submodules, namely, D ir2D with i s 0, 1. Moreo¨ er, if G is any G-in¨ ariant sublattice of D with the index Ž D : G . being a power of two, then there exists an integer k G 0 such that G g  2 kD , 2 kD0 , 2 kD 1 4 . Proof. Ž1. Again consider the sublattice L s DŽp . corresponding to the standard symplectic spread. Let H denote the subgroup of type SL 2 Ž q . ? 2 lying in AutŽp ., where q s p n Žsee the proof of Proposition 4.6.. It suffices to prove Lemma 5.9 for V2 viewed as H-module. Recall that L is a sublattice of odd index in D. So the index Ž D : Ž L q 2D .. is an odd n integer, which divides Ž D : 2D . s 2 p q1 . From this it follows that D s L q 2D, Dr2D s Ž L q 2D .r2D , LrŽ L l 2D .. But L l 2D = 2L, and Ž D : 2D . s Ž L : 2L .. Hence, in fact we have L l 2 D s 2L. We have shown that the F 2 H-modules Dr2D and Lr2L are isomorphic. Similarly, the C4 H-modules V4 s Dr4D and Lr4L are isomorphic. Therefore, in what follows we may suppose that V2 s Lr2L, V4 s Lr4L. In the proof of Proposition 4.6 we have chosen some orthogonal basis Žw z x < z g Fq j  `4. for L, and written down the action of H in this basis. We shall use the notation of that proof. Ž2. At this point we consider V2 . It is easy to see that the only nonzero K-stable vector in V2 is w s Ý z w z x. Here, the sum Ý z runs over all z g Fq j  `4 , and K s SL 2 Ž q .. Set U0 s  ¨ s Ý z a z w z x g V2 < Ý z a z s 04 , U1 s ² w : F 2 . Then 0 ; U1 ; U0 ; V2 is a composition series of the F 2 H-module V2 , with two trivial composition factors and one irreducible factor of dimension q y 1. The reduction modulo 2 of the scalar product Ž?, ? . provides an H-invariant nondegenerate symmetric bilinear form Ž?, ? . 2 on V2 . Clearly, U0 and U1 are dual to each other with respect to this form. Now let U be a nonzero proper H-submodule in V2 . Then dim U g  1, 2, q y 1, q4 . If dim U s 1, then U must be generated by a nonzero K-stable vector, that is, U s U1. If dim U s q, then the dual module U H has dimension 1; therefore U Hs U1 , which implies that U s U0 . Assume

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dim U s 2. Then the action of K on U induces a homomorphism from K to GLŽU . s GL2 Ž2. , S 3 . But K , SL 2 Ž q . is perfect; therefore this homomorphism is trivial; i.e., K acts trivially on U. In this case V2 has at least three Ždistinct. K-stable vectors, a contradiction. If dim U s q y 1, then the dual module U H has dimension 2, again a contradiction. We have shown that V2 has just two nontrivial proper submodules: U0 and U1. In particular, D0r2 D s U0 , D 1r2D s U1 , and D0 > D 1. Ž3. Next we consider any nontrivial proper submodule U in V4 , and suppose that 2V4 ­ U ­ 2V4 . Then ŽU q 2V4 .r2V4 is a nonzero submodule in V4r2V4 , V2 . By the results of Ž2., ŽU q 2V4 .r2V4 contains U1. From this it follows that U contains a vector ¨ s Ý z a z w z x with a z s "1. Here we are identifying the elements of C4 with 0, "1, 2. We also denote by Ž?, ? .4 the reduction modulo 4 of the scalar product Ž?, ? .. The operator Ry1 Žsee the proof of Proposition 4.6. sends ¨ to ¨ 9 s ya`w`x q Ý z g F q s a z wyz x. Setting U9 s U l 2V4 , we see that U9 contains u s ¨ y ¨ 9, with

Ž u, ¨ . 4 s 2 q

Ý

a z Ž a z y ayz . ' 2 Ž mod 4 . .

zgF q

In particular, U9 is a nonzero H-submodule in 2V4 . If U9 s 2V4 , then 2V4 : U, a contradiction. Therefore, by Ž2. we have U9 : 2U0 . But this last inclusion implies that Ž u, ¨ .4 s 0, a contradiction. This means: either U = 2V4 , or U : 2V4 . Ž4. Finally, let G be any G-invariant sublattice of D with Ž D : G . s m 2 . Then D = G = 2 mD. We prove by induction on m G 0 that there exists an integer k G 0 such that G g  2 kD, 2 kD0 , 2 kD 14 . This claim is obvious if m s 0 or 1 Žsee item Ž2... Now assume m G 2. Then D = G q 4D = 4D. Due to Ž3., either G q 4D : 2D, or G q 4D = 2D. In the former case, 2D = G = 2 mD; therefore D = 12 G = 2 my1D, and one can now use the induction hypothesis. In the latter case, 2 my 1D : 2 my 2 Ž G q 4D . s 2 my 2 G q 2 mD : G, and one can again use the induction hypothesis. Now we are in a position to prove Theorem 1.2. Consider any G-invariant lattice G lying in D. We may suppose that G ­ kD for any integer k ) 1. Clearly, G = lD for some natural l. Choose a minimal natural l with property G = lD. If l s 1, then G s D. Assume that l ) 1. Claim that l s 2 a p b for some nonnegative integers a, b. For, assume the contrary: l is divisible by an odd prime r, r / p. Observe that Ž G q rD .rrD is a nonzero H-module in Vr s DrrD. By Lemma 5.1, Ž G q rD .rrD s Vr , G q rD s D.

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Hence, l r

Ds

l r

l

Ž G q rD . s G q lD : G, r

contradicting the minimality of l. ˜ s G q p bD, one has Setting G

˜ = p bD , G ˜ = G = 2 a G q lD s 2 a G. ˜ D>G ˜ s f k Ž D .. Replacing G to fyk Ž G ., which is isometriBy Proposition 5.8, G ˜ s D. In this case, cally similar to G, we can suppose that k s 0, i.e., G D > G = 2 aD. By Lemma 5.9, G is similar to one of the lattices D, D0 , D 1. Theorem 1.2 has now been proved. When p s 3, Proposition 5.2 explains the presence of G-invariant lattices L in V with AutŽ L . s Ž C3 = Sp 2 nŽ3.. ? C2 predicted by w28x. Actually, from Proposition 5.2. Theorem 1.2, and the results of w28x we get the following consequence. COROLLARY 5.10. either

Let n G 3 and L be a G-in¨ ariant lattice in V. Then

Ži. AutŽ L . s G s Sp 2 nŽ p . ? 2; or Žii. p s 3, L is similar to D0 or D 1 , and AutŽ L . s Ž C3 = Sp 2 nŽ3.. ? C2 . The case n s 1 was also considered in w28x We emphasize that Corollary 5.10 is the only result in this paper, which uses the classification of finite simple groups. To conclude this section, we give the following concretization to the ‘‘even’’ part of Theorem 1.1. PROPOSITION 5.11.

Assume p ' y1 Žmod 8..

Ži. ŽNeighbour construction.. For any symplectic spread p of W s Fp2 n and a component L0 of p , set wqs wq Ž p . s

1 2

Ý ¨ Ž L. ,

Lg p

wys wy Ž p . s

1 2

Ý ¨ Ž L . y ¨ Ž L0 . .

Lg p

Then the lattices Dqs ² D0 , wq : Z , Dys ² D0 , wy : Z are Sp 2 nŽ p .-in¨ ariant e¨ en unimodular lattices. Žii. ŽModulo 2 prediction.. There exist Sp 2 nŽ p .-in¨ ariant sublattices D i , D > D i > 2D, i s 2, 3, such that the Ž1r'2 . D i are e¨ en unimodular.

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Proof. Ž1. We already know that D0 has index 4 in Ž D0 .*. At this point we show that wqq D0 , wyq D0 , and D _ D0 are three different nontrivial cosets in Ž D0 .*rD0 . For, let M be any Lagrangian. Then bL s Ž ¨ Ž L., ¨ Ž M .. are integer for all L g p . In particular, Ž ¨ Ž M ., 2 wq. s Ý L g p bL is integer. Furthermore, by Ž2. one has p ny 1 y Ž ¨ Ž M . , 2 wq . s

Ý

Lg p

bL2 y

Ý

Lg p

bL s

Ý

Lg p

bL Ž bL y 1 . g 2Z;

i.e., Ž ¨ Ž M ., 2 wq. is an odd integer. Therefore, Ž ¨ Ž M . y ¨ Ž M9., wq. g Z for arbitrary Lagrangians M, M9. Recall that D0 is generated by the vectors of form ¨ Ž M . y ¨ Ž M9.; hence we have just shown that wqg Ž D0 .*. As Ž wq, wq . s p Ž ny1.r2 Ž p n q 1.r4 is even, we are convinced that Dq is an even lattice containing D0 . Because Ž ¨ Ž M ., wq. f Z, wqf D. As a consequence, we get that Ž Dq : D0 . s 2 and so, Dq is unimodular. Also, wqq D0 / D_D0 . The same arguments can be applied to wy. If wqq D0 s wyq D0 , then ¨ Ž L0 . g D0 , contrary to the fact that D0 is even. Ž2. In order to prove assertion Ži., we need to show that Dq and Dy are Sp 2 nŽ p .-invariant. Consider any element g g G s Sp 2 nŽ p . ? 2. Then g fixes Ž D0 .*rD0 , and so g permutes three cosets wqq D0 , wyq D0 , D_D0 . The third coset is fixed by g, as D and D0 are g-stable. Therefore, in fact g permutes the two cosets wqq D0 , wyq D0 . We get a permutation representation of G of degree 2. But S s Sp 2 nŽ p . is the unique subgroup of index 2 in G, so S must leave each of these two cosets fixed. In other words, S leaves each of the lattices Dq, Dy fixed. ŽActually, this proof imitates well-known arguments with neighbours of a given lattice Žsee, e.g., the proof of Theorem 2.8 in w12x.. Therefore we have called Ži. neighbour construction.. Ž3. Now we prove assertion Žii.. Consider the F 2 S-module U s D0rD 1. Then it is known that U affords the Brauer character h q h , with h g IBr 2 Ž S . and QŽh . s QŽ yp .. Since p ' y1 Žmod 8., h and h can be written over F 2 . So we may suppose that U has some S-submodule U9, which affords S-character h. Fix a g g G_S and set U0 s g ŽU9.. Then U0 also is an S-submodule in U. If U0 l U9 / 0, then U0 s U9 and so U9 is a G-submodule in U, contrary to the fact that the G-module U is irreducible. Hence, U s U9 [ U0. Choose the sublattices D 2 , D 3 in D such that D 2 , D 3 > D 1 , D 2rD 1 s U9, D 3rD 1 s U0.

'

By their definition, D i Ž i s 2, 3. are S-invariant. We have to show that D i s 2DUi , and Ž u, u. g 4Z for any u g D i Žwhich mean precisely that Ž1r '2 . D i is an even unimodular lattice .. Recall that D 1 s 2Ž D0 .*. Therefore, the form Ž?, ? . 2 restricted to U is a nondegenerate S-invariant

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alternating bilinear form. But the character h is not self-dual, hence Ž?, ? . 2 restricted to U9 and U0 must be zero. This means that 2DUi = D i . On the other hand,

Ž 2DUi : 2D . s Ž DUi : D . s Ž DUi : D* . s Ž D : D i . s Ž D i : 2D . . Consequently, 2DUi s D i . It is now clear that Gi s  u g D i < Ž u, u. g 4Z4 is an S-invariant sublattice of index at most 2 in D i . Furthermore, Gi = 2D and Gi 2 2 wq. One can show that D 1 s ²2 wq, 2D : Z . So Gi = D 1. The irreducibility of U9 forces Gi s D i , as required. When n s 1, the F 2 S-modules D ir2D are known as extended quadratic residue binary codes of length p q 1. The mutual disposition of the lattices we have introduced is shown in the following diagram.

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6. SYMPLECTIC SPREADS AND CODES: n G 5 In this section we treat the case n G 5. Consider the Žunique. G-invariant odd unimodular lattice D s DŽ p, n. obtained by means of Theorem 3.6. Let

p s  Wi < 1 F i F p n q 1 4 be a symplectic spread of W s Fp2 n. Set ¨ i s ¨ ŽWi ., G s DŽp . s ² ¨ i < 1 F i F p n q 1: Z . For brevity we denote l s Ž n y 1.r2. Then G ; D s D* ; G* s pyl G. For each j, 1 F j F l q 1, one can view Hj s p jy1 G*rp j G* as the standard orthogonal space over Fp , with the basis Ž p jy1yl ¨ i < 1 F i F p n q 1. and with the form Ž x, y .Ž j. s p lq2y2 jŽ x, y . mod p. ŽHere and below, we identify the coset class x q p j G* with x.. Clearly, the H j ’s are isometric to each other, and so one can identify them canonically with H s Hlq1. Keeping this identification in mind, we can view every factor-group. C j s Ž Ž D l p jy1 G* . q p j G* . rp j G* as a linear code of length p n q 1 over Fp , with ambient space H. It is obvious that C1 : C2 : ??? : Cl . PROPOSITION 6.1. Under the abo¨ e assumptions, one has C jHs Clq1yj for 1 F j F l. In particular, C j is self-orthogonal if 1 F j F Ž l q 1.r2; and CŽ nq1.r4 is self-dual if n ' 3 Žmod 4.. Proof. We find the orthogonal complement to C j in Hj , H. A vector x g p jy1 G* belongs to this complement if and only if p lq2y2 j Ž x, Ž D l p jy1 G*. q p j G*. : pZ, i.e., Ž x, Ž D l p jy1 G*. q p j G*. : p 2 jy1yl Z. This last condition is equivalent to that p lq1y2 j x belongs to

Ž p j G* . * l Ž D* q Ž p jy1 G* . * . s p lyj G* l Ž D q p lyjq1 G* . s Ž D l p lyj G* . q p lq1yj G*. It remains to recall that the identification of H j with Hlq1yj is actually performed by multiplying to the scalar p lq1y2 j. EXAMPLE 6.2. Assume n ' 3 Žmod 4.. Then the odd unimodular lattice D provides us at least two self-dual codes of length p n q 1 over Fp . Namely, we take p to be either the standard symplectic spread, or the symplectic spread constructed by Bader, Kantor, and Lunardon w4x, and consider the corresponding ‘‘middle’’ code CŽ nq1.r4 . It seems that in the

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former case we get the middle quadratic residue code described by Ward w33x Žsee also Conjecture 6.3 below.. In the latter case we get another code, which is presumably new. From now on to the end of the section we take p to be the standard symplectic spread. Then the same arguments as in the proof of Proposition 4.6 assure that all the codes C j , 1 F j F l, are among the GL2 Ž q .-codes ha¨ ing a Fp-form, which ha¨ e been introduced by Ward in w33x. He has shown that the lattice of his GL2 Ž q .-codes is inversely isomorphic to the lattice of the so-called closed subsets of F 2n. He has also constructed the following analogues of Reed]Muller codes. View elements of F 2n as binary words of length n and take Bw to be the set of all binary words of length n and weight F w. Then Bw is closed and cyclic Žin the sense of w33x., and Ward’s correspondence gives us a GL2 Ž q .-code Cn, w over Fp , 0 F w F n y 1. The middle code is just Cn, Ž ny1.r2 ; more generally, Cn,Hw s Cn, ny1yw . The following conjecture seems to be very plausible. Conjecture 6.3. Under our hypotheses, one has C j s Cn, ny2 j for j, 1 F j F l s Ž n y 1.r2. This conjecture is obviously compatible with Proposition 6.1. Furthermore, it is true for n s 3 Žsee Proposition 4.6.. Moreover, it gives us a good estimate for the minimum of the lattice D. PROPOSITION 6.4. Ži. p

Assume that Conjecture 6.3 is true. Then

Ž ny1.r2

q1

2 Žii.

F min D F p Ž ny1.r2 .

Assume that n G 3. Then Ž p q 1.r2 F min D F p Ž ny1.r2 .

Proof. Ži. It is clear that min D F p l. In order to prove min D G Ž p l q . 1 r2, it is enough to accept Conjecture 6.3 and show that min Cn , w G

p wq 1 q 3

Ž 6.

2

for 0 F w F n y 1. ŽIn fact, suppose that Conjecture 6.3 is true and Ž6. has been established, but a nonzero vector ¨ g D has the norm Ž ¨ , ¨ . F Ž p l y 1.r2. Then it is easy to see that ¨ g G*_p l G*. In this case, one can find an index j, 1 F j F l such that ¨ g p jy1 G*_p j G*. This implies that

Ž ¨ , ¨ . G min Cn , ny2 j ? Ž p jy1yl ¨ 1 , p jy1yl ¨ 1 . G a contradiction..

p nq 1y2 j q 3 2

? p 2 jy2yl )

pl 2

,

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151

Next we proceed to prove inequality Ž6.. It was already established for the cases w s n y 1, Ž n y 1.r2 in w33x. Therefore we can suppose that 0 F w F n y 2. Following w33x, we represent the exponents of the code C s Cn, w as numbers written base p with n digits. Set h s Ž p y 1.r2. Claim that all numbers r in the range from 0 to p wq 1 y 1

00 ??? 00^hh` ??? hh _s

2

wq1

are exponents for C0, the twice punctured code corresponding to C. In the terminology of Ward’s correspondence, our claim is equivalent to saying that, for any such an r, r q r 9 has at most w ‘‘carries.’’ Here, r 9 s Ž p n y 1.r2 y r. For, we can write such r in the form r s ^`_ 0 ??? 0 a wq 1 a w ??? a1 ny 1y w

with 0 F a i F p y 1. Furthermore, p n y p wq 1

r9 s

2

q

ž

p wq 1 y 1 2

y r s ^` h ??? h _bwq 1 bw ??? b1

/

ny 1y w

with 0 F bi F p y 1. As a i q bi F 2 p y 2 and h q 1 s Ž p q 1.r2 F p y 1, there are no carries in r q r 9 from the positions w q 2, w q 3, . . . , n. Suppose that a wq 1 q bwq1 G h q 1. Observe that h ??? h 0 ??? 0 s Ž p ^` _^`_

n

y p wq 1 . r2.

ny 1y w wq 1

So we have pn y 1 2

s r q r9 G

p n y p wq 1 2

q p w Ž h q 1. s

pn y 1 2

q

pw q 1 2

,

a contradiction. This means that r q r 9 cannot have a carry from the position w q 1. Hence, r q r 9 has at most w carries: these carries may occur only from positions 1, 2, . . . , w. ŽRemark that s q s9 has exactly w q 1 carries if s s Ž p wq 1 q 1.r2.. Now applying the BCH bound w22x to the cyclic code C0, one sees that min C0 G Ž p wq 1 y 1.r2. The double transitivity of GL2 Ž q . implies that C s Cn, w has minimum weight at least Ž p wq 1 q 3.r2.

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Žii. First we prove the following assertion: Let C be any of Ward’s codes w33x of length p n q 1. Assume that n is odd, C has Fp-form, and C is self-orthogonal. Then min C G

p Ž nq1.r2 q 3

. Ž 7. 2 For, consider the subset B of 2 V assigned to C by Ward’s correspondence, where V s  1, 2, . . . , n4 . Then the fact C has Fp-form is equivalent to that B is closed under the cyclic permutation i ¬ i q 1 Žmod n. of V. Furthermore, the self-orthogonality of C is equivalent to the following condition: for any B : V, either B g B, or V_B g B. These observations allow us to see that B0 s  1, 2, . . . , Ž n y 1.r24 and all its subsets belong to B. In this case, the same arguments as in item Ži. show that all numbers r in the range from 0 to 00 ??? 00^hh` ??? hh _s Ž n q 1 . r2

p Ž nq1.r2 y 1 2

are exponents for C 0, the twice punctured code corresponding to C . Again applying the BCH bound, we get estimate Ž7.. Now suppose that a nonzero vector ¨ g D has the norm Ž ¨ , ¨ . F Ž p y 1.r2. Then it is easy to see that ¨ g G*_p l G*. In this case, one can find an index j, 1 F j F l, such that ¨ g p jy1 G*_p j G*. This implies that

Ž ¨ , ¨ . G min C j ? Ž p jy1yl ¨ 1 , p jy1yl ¨ 1 . s min C j ? p 2 jy2yl . If 1 F j F Ž l q 1.r2, then C j is self-orthogonal by Proposition 6.1. By Ž7., min C j G Ž p Ž nq1.r2 q 3.r2 s Ž p lq1 q 3.r2; hence Ž ¨ , ¨ . ) p 2 jy1r2 G pr2, a contradiction. If j G Ž l q 2.r2, then min C j G min Cn, 0 , since all nontrivial Ward’s codes are contained in Cn, 0 Žcf. w33x.. Applying Ž6., one gets min Cn, 0 G Ž p q 3.r2, and so Ž ¨ , ¨ . G Ž p q 3.r2, again a contradiction. EXAMPLE 6.5. Assume n s 5. It is not difficult to see that F 25 has just nine cyclic closed Žproper. subsets; namely, Bw with 0FwF4 and Bw, k with ws2, 3, ks0, 1. Here, b 3, 0 contains all binary words of weight F 2, together with the five words 11100, 01110, 00111, 10011, and 11001. B3, 1 contains all binary words of weight F 2, together with the five words 11010, 01101, 10110, 01011, and 10101. Finally, B2 , 0 s  00000, 10000, 01000, 00100, 00010, 00001, 11000, 01100, 00110, 00011, 10001 4 , B2 , 1 s  00000, 10000, 01000, 00100, 00010, 00001, 10100, 01010, 00101, 10010, 01001 4 .

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Ward’s correspondence gives us nine GL2 Ž p 5 .-codes of length p 5 q 1 over Fp : C5, w with 0 F w F 4, and C5, w, k with w s 2, 3, and k s 0, 1. Furthermore, C5,Hw s C5, 4yw , and C5,Hw, k s C5, 5yw, 1yk . Thus mutual disposition of these codes is as in the diagram below:

By Ž6. one has min C5, w G Ž p wq 1 q 3.r2. Furthermore, in the proof of Proposition 6.4 we have shown that: if 0 F r F Ž p 4 y 1.r2, then r q r 9 has at most three carries; moreover, if r q r 9 has exactly three carries, then these carries occur from the positions 1, 2, and 3. In other words, all the numbers in the range from 0 to Ž p 4 y 1.r2 are exponents for Žthe twice punctured code corresponding to. C5, 3, 0 . By the BCH bound, min C5, 3, 0 G Ž p 4 q 3.r2. Similarly, min C5, 2, 0 G Ž p 3 q 3.r2. As C5, 2, 1 : C5, 1 , min C5, 2, 1 G Ž p 2 q 3.r2. Finally, min C5, 3, 1 G Ž p 3 q 3.r2, since C5, 3, 1 : C5, 2 .

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We were not able to identify exactly the codes C1 , C2 related to our lattice D and the standard symplectic spread among these nine codes. Here are some partial arguments. First we know that C1 : C2 s C1H . Furthermore, from the proof of Proposition 3.4 one can extract that C1 has a codeword of weight p 4 . So C1 cannot be C5, 4 Žbecause of the lower bound for the minimum weight., C5, 0 , C5, 1 , C5, 2, 0 , C5, 2, 1 Žbecause of the self-orthogonality.. By Proposition 6.4, min D G Ž p q 1.r2. Now assume that C1 is neither C5, 3, 1 nor C5, 2 . Then C1 is either C5, 3, 0 or C5, 3 . ŽDue to Conjecture 6.3, C1 s C5, 3 .. Then from the lower bound for minimum weights given above we immediately get min D G Ž p 2 q 1.r2. Remark 6.6. Fix a prime p with p ' 3 Žmod 4.. In a sense, one can consider our Sp 2 nŽ p . ? 2-invariant lattices of rank p n q 1, unimodular if n is odd and p-modular if n is even Žthe latter will be investigated in our subsequent paper., as ‘‘ p-analogues’’ of the Barnes]Wall lattices BW2 n w3x. It is known that these lattices have rank 2 n, minimum 2 wŽ ny1.r2x. Furthermore, they are unimodular if n is odd, and 2-modular if n is even. Finally, 1q 2 n 1q Ž2 ny2. Aut Ž BW2 n . s 2q ( Vq ( Vy 2 n Ž 2 . > 2y 2 ny2 Ž 2 . .

Modulo Conjecture 6.3, our lattice behave themselves like the Barnes]Wall lattices, in the manner of their minimum and of their construction as well. This observation is consonant with the similarity between the groups 1q Ž2 ny2. Ž . Ž . 2y ( Vy 2 ny2 2 and Sp 2 n p from the point of view of the global irreducibility of certain of their representations w15x.

ACKNOWLEDGMENTS The authors are grateful to Professor W. M. Kantor and Professor G. Lunardon for helpful discussions on the subject of symplectic spreads. The second author is grateful to Professor Dr. G. O. Michler and his colleagues at the Institute for Experimental Mathematics for stimulating conversations and their generous hospitality. Finally, they are indebted to the referee who made some suggestions for improvements.

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