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Center sets and ternary codes
JOURNAL OF ALGEBRA 65, 206--224
(1980)
Center Sets and Ternary Codes HAROLD N. WARD* Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903 Communicated by Walter Felt Received September 3, 1979 TO H . W .
BRINKMANN, ON THE OCCASION OF
HIS EIGHTIETH BIRTHDAY
INTRODUCTION In recent years the classification of self-dual codes has been carried almost to the limits on lengths beyond which the number of codes becomes overwhelming. The survey article by Sloane [15] refers to most of the papers making up this project; and the work by Pless, Sloane, and me [13], for which the present paper provides supporting material, brings his bibliography up to date. The approach used in that joint paper, which deals with codes over GF(3) or "ternary" codes, is typical of the classification for other fields. Let me give it a brief sketch. Decomposable codes of a given length are by definition direct sums of shorter codes and so determined by previous classifications. If an indecomposable code contains a fair number of words of weight 3 (all the words in a self-dual ternary code have weights divisible by 3), it will contain a large known decomposable subcodc. Additional codewords then serve to "glue" the components of this subcode together. The entire code can be constructed by a systematic study of possible glue words. (There is a good exposition of the glueing strategy in the paper of Conway, Pless, and Sloane [3].) When there are few or no words of weight 3, finding suitable decomposable subcodes can be difficult. One tries to construct the codes themselves by any means, fair or foul. At the same time the group of each code in hand is worked out so that the number of codes equivalent to it is known. Because the entire number of codes sought can be predicted, one knows when to stop. In fact, the nature of the difference when one is still short of the total can serve as a clue to the automorphism structure of missing codes (this is nicely illustrated by Conway and Pless [2]). Consider now a self-orthogonal ternary code having minimum weight 6, the * This research was partially supported by National Science Foundation Grant MCS 78-01458. 206 0021-8693/801070206-19502.00/0 Copyright © 1980 by Academic Press, Inc.
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largest possible minimum weight for self-dual codes of lengths at most 20. Each word of weight 6 determines a "hexad", the binary word obtained by changing any - - l ' s to l's and pretending the result has GF(2) entries. These hexads have the property that the sum, over GF(2), is again a hexad if and only if they share three l's; or, what turns out the same, if and only if their dot product is 1. They form an example of what is called a "center set". The combinatorial aspects of center sets restrict the possible disposition of the hexads enough to facilitate significantly the construction of codes. This technique was employed at the "fair or foul" step of our paper. Section 1 of the present paper contains the definition and basic properties of center sets. Each center set involves a graph in a natural way, and the first goal of the paper is the description of center sets with connected graphs. A review of properties of binary quadratic spaces and the construction of one class of center sets occupies Section 2. McLaughlin's classification of irreducible groups generated by transvections [11] provides the tool used in Sections 3 and 4 to classify connected center sets. Section 5 links ternary self-orthogonal codes and center sets. The major result is that the center sets arising are of symmetric type; that is, they involve the symmetric group in McLaughlin's classification. With this section the background for the application to self-dual codes of length 20 in our paper is complete. In that paper the reader will find examples of code constructions making use of the detailed descriptions of center sets to arrive at generating matrices. In Section 6 of this paper, however, I have given a construction of the unique self-dual ternary code of length 16 and minimum weight 6 to illustrate the kind of insight the center set approach provides. Part of the point is to give an example of a connection between ternary and binary codes that is a consequence of the center set perspective. The combinatorial structure of the hexads for this particular code leads to a binary self-dual code of length 32 and minimum weight 8, and the details are in the last section of the paper. The code constructed is different from two well-known examples, the Reed-Muller code and the extended quadratic residue code. Another instance of this kind of connection is that between the ternary extended Golay code of length 12 and the binary extended Golay code of length 24. One view of this was given by Rasala [14]; the view using center sets is set out in my Coxeter Symposium paper [17]. For background material in coding theory the reader should consult the book by MacWilliams and Sloane [9], to which several explicit references are made later on. In this paper the dimension of a vector space V is denoted dim V, and the subspace spanned by a set X is ( X ) . Displays of row vectors or matrices have blank spaces for zeroes except perhaps for selected positions used to clarify the pattern.
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HAROLD N. WARD 1. CENTER SETS
Center sets are certain subsets of symplectic vector spaces of finite dimension over the binary field GF(2). Let V be such a space and let ~bbe the corresponding symplectic form. It may be that ~b is degenerate; its radical is given b y rad~ ={vcV
I~(u,v) =0foralluin
V}.
(Standard references for symplectic spaces as well as for the material on quadratic spaces needed in Section 2 are the books of Dickson and Dieudonn4 [4, 6, 7].) DEFINITION |.1. A center set is a nonempty subset J of a symplectic space over GF(2) with this property: whenever u and v belong to J, u q- v is in J if and only if ~b(u, v) ~ 1. T h e members of J can be taken as the vertices of an undirected graph F in which two vertices u and v are joined by an edge exactly when ~b(u, v) - - 1. T h u s if u and v are adjacent (connected by an edge), u, v, and u ~ v will form a triangle. J also produces a linear group L on the vector space V containing J , generated by the transvections
tu: v --~ v + ~(v, U)U for each u in J - rad ~b (the "transvections" obtained from the u in rad ~b are each the identity). PROPOSITION 1.2. Let J be a center set and L its group. Then L preserves J, and the L-orbits of J are exactly the vertex sets of the connected components of the graph F of J. Each component has diameter at most 2.
Proof. If u ~ J - rad ~b, v ~ jr, and t~(v) ~ v, then ~b(u, v) - - 1 and tu(V ) = u ~ v, a m e m b e r of jr. L consequently preserves J ; and since ~b(u @ v, v) - - 1, u @ v is in the connected component containing v. T h u s the L-orbit of v is in that component. On the other hand, if w is adjacent to v, v @ w is in J and w = tv+w(V). It follows that each component is a n L - o r b i t . F o r the diameter statement, note that if u, x, y, and v are successively adjacent members of J for which u and y are not adjacent and x and v are not adjacent, then x @ y is in J and is adjacent to u and v. So any path joining two members of J can be modified step-by-step until it has length at most 2. W h e n J is a connected center set (one whose graph is connected) the sets {t~rad¢ I v @ t~J}, determined by each v in J , will not depend on v because of the transitivity of L on J and the fact that L acts trivially on rad ~b. Consequently the c o m m o n set T is a subspace of tad ~b. F o r if t 1 and t 2 are in T and v ~ jr, then v -t- tl ~ J and
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then (v + tl) + t 2 e J also. That is, v + (q + t2) e J and t 1 + t 2 ~ T. Of course 0 E T; incidentally, 0 is never in a center set. DEFINITION 1.3. Let J be a connected center set. The kernel of J is the subspace ker J, where kerJ={t~rad¢[v+t~JforeachvinJ}. It will appear later that when J spans V, ker J is most of rad ¢. PROPOSITION 1.4. Let J be a connected center set spanning the corresponding space V. I f t: v ~ . v + ¢(~, u)u
is a transvection in the group L of J, then the center u of t belongs to J. Proof. By convention u 6 rad ¢, so that t(v) ~ v for some v in J, as J spans V. T h e n t ( v ) = u + v and u + v ~ J. But then u = ( u + v ) + v a n d u c J . I f J i n the proposition is not trivial, that is, not just a singleton set, J c~ rad ¢ = ~ , since something in rad ¢ would not be adjacent to anything else. The proposition says that J is exactly the set of centers of the transvections of L. This is the reason for using the name "center set". Let V' -- V/rad ¢, with the accent also denoting the quotient mapping. The form ¢' induced by ¢ on V' is nonsingular, as long as ¢ is not identically 0. L acts on V', producing theilinear group L' generated by the transvections with centers in J ' . PROPOSITION 1.5. I f J is a connected center set spanning V, then L' acts irreducibly on V'. Moreover, J ' is the set of centers of the transvections in L'. Proof. An L'-invariant subspace of V' is the image W' of an L-invariant subspace W of V containing rad ¢. If w c W - - rad ¢, then for some v in J, ¢(% w) = 1. Then v + w = t~(w), so that v - / w is in W, too. Thus v ~ W, and by the transitivity of L on J, all of J i s in W a n d W = V. Now J ' may not be a center set, but at least it has the property that if u' and v' are in J ' and ¢'(u', v') 1, then u' + v' 6 J ' . T h e second statement of the proposition follows just as in the proof of Proposition 1.4.
2. QUADRATIC SPACES Proposition 1.5 points to the use of McLaughlin's description of irreducible groups generated by transvections [11] in classifying connected center sets. Preparatory to that, let me summarize some properties of quadratic spaces over GF(2).
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On a finite-dimensional vector space V over GF(2), a quadratic form whose associated symplectic form is ~b is a function Q: V ~ GF(2) for which
Q(u + v) = Q(u) + Q(v) + 4J(u, v) for all u and v in V. A subspace and the quadratic form induced on it by restriction of Q are called singular if ~b is degenerate there, and totally singular if Q vanishes on the subspace. A nonsingular plane (two-dimensional subspace) may have just one vector on which Q is 1, in which case the plane is hyperbolic; if not, Q will have the value 1 on all three nonzero vectors and the plane is elliptic. Any quadratic space (one endowed with a quadratic form) that is nonsingular is an orthogonal sum of elliptic and hyperbolic planes. It is completely determined up to isometry by its dimension 2n and its index, the common dimension of its maximal totally singular subspaces. That index is n or n -- I according as the number of elliptic planes, in an orthogonal decomposition into planes, is even or odd. If the quadratic space V is singular, choose a supplement V0 to rad ~b, so that V is the orthogonal sum Vo ± rad ~b. T h e n the quadratic form Q is given by
Q(o + r) = Qo(~) + A(r) for v in Vo and r in rad ~b. Here Qo is the restriction of Q to Vo and nonsingular, while A is the restriction to tad ~b and actually a linear functional. When A = 0, Qo can legitimately be thought of as the form induced by Q on V/rad ~b; but when A 4 = 0, proper choice of Vo allows Qo to be taken with either of the two possible indices (provided Vo =/= 0). An important theorem is the one of Witt (which has an analogue for symplectic spaces): if V is a nonsingular quadratic space, any isometry between two subspaces can be extended to one of V. T h e appeals to Witt's theorem in the sequel involve constructing from a given spanning set of V a subspace isometric to the particular one being dealt with at the time. That the needed constructions can be made will usually be taken as evident. T h e results that follow illustrate the preceding ideas; Vo and A will continue to be used as above. PROPOSITION 2.1. Let V be a finite-dimensional quadratic space over GF(2), with quadratic form Q and associated symplectic form ~b. Then if the set J is nonempty, where
j = {v ~ V]Q(v) = 1 and v • rad ~b), it is a center set. Proof. Saying that J is nonempty is the same as saying rad ~b 4: V, that is, ~bis not identically 0. If u and v are in J, Q(u + v) -~ ~b(u, v). Thus if u + v E J,
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~b(u, v) = 1. A n d if ~b(u, v) = 1, u + v e J provided u + v q~rad ~b. But since ~b(u, u + v) = ~b(u, v) =/= 0, this is so. PROPOSITION 2.2. Suppose that J in Proposition 2.1 is nonempty. Then ( J ) - V unless V o can be taken as a hyperbolic plane. Proof. If V o is the hyperbolic plane (u, v), with Q(u) and Q(v) both 0, then J is the union of the sets u + (rad ~b - - her A), v + (rad ~b - - her A), and u + v ~- her A (the first two may be empty). T h u s J does not span V. On the other hand suppose V 0 cannot be arranged as a hyperbolic plane, and let u ¢ rad ~b and u ¢ J . T h e n Q(u) = 0, and one may choose V o to have u in it. By W i t t ' s theorem there is a hyperbolic plane (u, v ) in V 0 . Because (u, v ) =/= Vo, there is a m e m b e r w of V o for which Q(w) 1 and w c (u, v ) ± (if V o is written as an orthogonal sum of planes one of which is (u, v), w may be selected from one of the other terms). T h e n w ~ J and u + w e J, so that u e ( J ) . ( j ) therefore contains all of V - - tad ~b; as that is more than half of V, ( J ) = V. Evidently J would still be a center set if the excluded members of rad ~b were admitted. T h e trouble is that any such elements will form isolated components in the graph. T h e next proposition settles connectivity for the original J. PROPOSITION 2.3. I f J is a nonempty center set of the kind in Proposition 2.1, then J is connected unless h = 0 and either V o is a hyperbolic plane and rad ~b 4 : 0 or else V o has dimension 4 and index 2. Proof. First suppose A @ 0. Let ./'1 and J2 be two distinct members of J, where j~ = vi + r i , vi E V0, and ri ~ rad ~b. In the verification that Jl and J2 are in the same connected component it may be assumed that ~b(j1 ,J2) ~ 0, that is, that ~b(vl, v~) = 0. If v 1 = v2, take v in V 0 for which ~b(v~, v) = 1 ; and if v~ ~ v2, use W i t t ' s theorem to produce a v in V o for which ~b(Vl, v) = ~b(vz, v) = 1. (In the second instance @1, v~) is isotropic and 2-dimensional.) T h e n take j to be v 4- r with r chosen in rad ~b to put j in J ; Q(r) = 1 + Q(v) and of course v 4- 0. T h e equations 4~(jl , j ) = 4J(j2 , j ) = 1 show Jl and J2 to be in the same component. N o w let A = 0. I f V o is a hyperbolic plane (u, v ) as in Proposition 2.2, J is the coset u + v + rad ~b and is totally disconnected when not a singleton set. I f V 0 = E1 5_ E2, the orthogonal sum of two elliptic planes, J is the union of the six cosets v + t a d ~b, where v =/= 0 and either v E E1 or v e E~. T h i s time J breaks into two components. I n any but these cases the argument is as before: again letj~ and j2 be distinct members of J , where ~b(j~ ,J2) = 0 and j i = vi + r , . I f V 1 = ' U 2 , select j in Vo making @1 , J ) an elliptic plane (so Q ( j ) = 1). I f v 1 4 v2, take u 1 and u 2 in V o making @1, u l ) elliptic, @2, u2) hyperbolic, and the two planes orthogonal. T h e n let j be u 1 + u 2 . T h e troublesome possibilities for Q0 for the application
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of Witt's theorem are those discussed above. In either situation, j ~ J ~ ( J l , J ) == ~(J2,J) -- 1, establishing the result. T h e proof of the last item of this section is routine: PROPOSITION 2.4.
and
For the center set of Proposition 2.1, the kernel is ker A.
3.
CLASSIFYING
CENTER
SETS
Let J be a connected center set spanning its vector space V. Assume that J is not trivially a singleton set, so that V 4: rad 4. By Proposition 1.5 the group L ' acts irreducibly on V'. L' is generated by the transvections with centers in J ' and J ' is the set of centers of the transvections in L'. T h e work of McLaughlin [11] yields these possibilities: (1) Symplectic type: L' is the symplectic group of the form ¢' induced by ¢ on V', and J ' is the set of all nonzero members of V'. (2) Quadratic t y p e : L ' is the orthogonal group of a quadratic form O' on V' having 4' as its associated symplectic form. T h e dimension of V' is at least 4, but the case that dim V' = 4 and O' has index 2 is missing. J' is the set of members of V' on which O' has value 1. T h e possibility that dim V' = 6 and Q' has index 3 will also be ruled out (and listed with the next type) for reasons that will become apparent. (3) Symmetric type: L' is isomorphic to the symmetric group on at least five letters. T h e details of this type will be given in Section 4. T h e types in this classification will be ascribed to the center sets themselves. T h e center sets of Section 2 account for the symplectic and quadratic types, principally because of the next result. LEMMA 3.1. Let J be a nontrivial connected center set spanning its space V, so that V :/= rad ¢. Suppose J is of symplectic or quadratic type. If j l , J2 , and j~ are three mutually orthogonal members of J and Jl + J2 + J3 ~ rad 4, then in fact
Jl + J2 + J~ e J. .t
.t
.t
Proof. In V', Jl + J~ + J~ =# 0. If the Ji are dependent, they might all coincide. In that case take j in J to satisfy ¢'(j~, j ' ) = 1, with (j~, j ' ) an elliptic plane when J is of quadratic type. If thej~ are dependent but not all coincident, •t .t ! t it must be that on renumbering j'~ 7~ J2 but j~ = j~. T h e n take u 1 and u2 so that ! / .t .t .t (J1, ul) and \J2, u~) are orthogonal nonsingular planes; (J1,32) is an isotropic t • .t t plane. This time if J is quadratic make (j~, ut) elllptm and (J2, u2) hyperbolic. T h e only block to doing that is the situation that dim V' = 4 and Q' has index 2, the first excluded possibility in the quadratic case. Then u'1 + u'2 c J ' because Q'(u~) = 1 and Q'(u~) = 0. L e t j be a preimage of u~ + u'2 in J. .t
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If the j'£ are independent, construct three mutually orthogonal nonsingular .t t planes ( J i , ui), each one elliptic in the quadratic case; that involves the second exclusion. Again u'1 + u'2 @ u'a ~ J ' and j is taken in J satisfying j ' = u'1 +
u'2 + u'3.
For all possibilities the j produced is in J and ~(j, ji) = I for i = 1, 2, 3. Then, successively, j, j + j l , J+Jl+J2, J+JI+L+J3, and finally j~ @ jz + ja are all in J. Notice that if J is quadratic, j~ + j2 + J ~ ~ rad ~b automatically, because Q'(jl + J~ + J'3) = 1. And the conclusion of the last paragraph is of some use by itself: COROLLARY 3.2. t° Jl , J2 , and ja .
With the assumptions of Lemma 3.1, there is a j in J adjacent
THEOREM 3.3. Let J be a connected center set spanning its space V, where V @ rad ~b. Then i f J i s ofsymplectic or quadratic type, it is one of the sets described in Proposition 2.1. The quadratic form producing J is given by Q(v) = 1 if v ~ j or v e rad ~b -- ker J, O(v) = 0 otherwise. Proof. As the formula suggests, the proof amounts to showing that Q is really a quadratic form having ~b as its associated symplectic form. Because J spans V, it is enough to verify that Q(v + j) = Q(v) + Q(j) + ~b(v,j) for all v in V and j in J. If v a J the equation is Q(v + j) = ~b(v,j) and its correctness is almost just the definition of a center set. However, one needs to observe that if ~(v, j) = 0, then v + j is in ker J if it is in rad ~b, since its sum with j is back in J. When v E rad ~b, the equation becomes Q(v + j) = Q(v) -? 1 and it is a recasting of Definition 3.1 for ker J. Suppose then v is neither in J nor in rad ~b. Write v as a sum of members of J using as few as possible. If two terms were not orthogonal their sum would be in J and the number of terms could be reduced; so the terms are orthogonal. T h e n if there were three or more terms, L e m m a 3.1 would imply that the sum of any three of them is in tad ~b, otherwise the sum could again be shortened. As v q~rad ~b, there could not be exactly three terms; but if there were more than three, the sums of pairs (as sums of overlapping groups of three) would also be in rad q~. A little thought reveals that v would end up in rad ~b. Consequently v is the sum jl +/'2 of two orthogonal members of J. If ~b(v,j) ~ 1, then ~b(jl,j) = 1 and ~b(j2,j) = 0 , say. Thus v + j =
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(/'1 + J) + / ' 2 , a sum of two orthogonal members of J. So v ~- j is in ker J if it is in tad ¢. Hence Q(v -k j ) = Q(v) = o and the equation is correct. If ¢(% j) = 0 and both ¢(Jl ,J) and ¢(/'2 ,J) are 0, then as v j - j ----/'1 -kj~ + j , L e m m a 3.1 implies that v + j is either in J or tad ¢. But if it is in rad ¢, it is not in ker J, because it plus j is not in J. This time, then, Q(v) = 0 and Q(v -[- j) = 1 as needed. Finally, if ¢(/'1, J) -- ¢(J~, J) - 1, Jl + J and then/'1 +/'2 -k j are in J and the equation is again correct.
4. SYMMETRIC CENTER SETS For a center set of symmetric type, the g r o u p L ' is isomorphic to the symmetric group on m letters for a certain m. I n the division of types in Section 3, m > / 5 . It is profitable to relax that restriction, even though there is some duplication with the other types (which will be discussed later). Henceforth a symmetric center set will mean one in which L' is isomorphic to a symmetric group acting on V' in the manner about to be described, with the corresponding J ' . Let S be the symmetric group on m letters, the "letters" being 1, 2,..., m. Let GF(2) m be the space of m-tuples over GF(2), with standard basis e1 ..... era; e~ is the m-tuple with a 1 in the i-th place and O's elsewhere. T h e action of S on GF(2) m is given by
g(al ..... am) = (ag-~(1) ,..., ag-l(m)) when g ff Sin. Thus g(ei) = eg(i) , S preserves the standard inner product on GF(2) m which is the sum of the component products. It also preserves the even subspace E of m-tuples having component sum 0, On E the inner product is a symplectic form ¢, and the radical of ¢ is 0 if m is odd, but the span of the all-1 m-tuple if m is even. Let * stand for the map of E onto the quotient E* of E by the radical of ¢, and let ¢* be the induced form on E*; it is nondegenerate as long as m ) 3. For a symmetric center set, V' is isometric to E* endowed with the form ¢*, for an appropriate m. Furthermore, if P is the set of pair sums ea -? eb, where a v~ b, J ' under the isometry is P*. The representation of S involved here was studied by Dickson [5]. This description of V' leads in turn to one of V and J ; the discussion that follows will be summed up in Theorem 4.1. When m is odd, identify V' with E .t and J ' with P. Take Jla in J so that Jla = el + ca, where a > 1. T h e n when •! .t a > I, b > 1, and a @ b , it follows that ¢(Jla,Jlb) = ¢ ( J l a , J l ~ ) = 1 and jl~ -? Jib is in J. Put
L~ =Ao +Ab. From this, j~b -----ea -}- eb, and if Vo is the span of the jab, Vo will be isometric
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to E and V = V o 5_ rad ~b. T h e members of J are the members of the cosets
jab + ker J , and since J is supposed to span V, ker J = rad ~b. T h u s when m is odd, V can be taken as E _[_ R, the form ¢ being 6 on E and 0 on R. T h e kernel of J is R and J is the union of the cosets p + R, where p E P. Notice that the transvection t , on E coming from a pair sum p in P is the transformation produced by the transposition (ab) in S, where p --~ ea + e~. N o w suppose m is even. Let e ~ e1 + ' " + e m be the all-1 m-tuple, so that r a d ¢ = (e). V' is to be identified with E* and J ' with P*; m ~ 4. Again select Jla in J for which J~a ~- e* + e* and let ja~ = ]la + jlb" I f ker J = o, v is spanned by the jab and the kernel of the h o m o m o r p h i s m of V onto E* is (J12 + "'" + jim). I f this is 0, V can be taken as E*, and if not, as E. J is P* or P in the two cases. I f ker J is not 0, there is some flexibility in the choices of thejaa's, and one can make sure that if w : Jl~ + "'" + J l - , , w :/: 0. T h e n if V 0 is the span of the jab , Vo is isometric to E. Because J is the union of the cosets jab + ker J , v = v o+kerJ and V ~ V 0 _ l _ R for some subspaee R of k e r J . I n addition, rad ¢ : ( w ) _1_ R. T h e sum w may or m a y not be in ker J . W h e t h e r ker J is 0 or not, when m : 6 the sum J12 + J13 -]- "'" + Jl~ cannot be in k e r J ; in particular it is not 0. F o r that sum is j l ~ +J84 +J56; were it in ker J , its sum j8 a -+-J56 with J12 would be in J . But ¢(J34, J56) -/: 1. W h e n m : 3 or m : 4 the center set is also of symplectic type, with dim V' : 2. T h u s one can also assume m 4- 4 (the set-up for 3 is easier to visualize!) T h e final s u m m a r y of symmetric center sets is this: THEOREM 4.1. Let J be a nontrivial connected center set of symmetric type, spanning its vector space V. Then up to isometry V and J are one of the following arrangements, with the notation as above. R is a finite-dimensional GF(2)-space carrying the 0 form. (i)
V=E_l_R,J={p+rlpeP,
(ii) V = E ± R , J = { p + r , p + e + r l p e P , and m is even and at least 8. (iii)
rER};kerJ=R,
andm>/3, m~4.
r~R};kerJ=(e)+R,
If = E*, J = P*; ker J = 0 and again m is even and at least 8.
Verifying that these descriptions do give connected center sets of symmetric type is straight-forward. F o r applications the parameters of the corresponding graphs need to be given. COROLLARY 4.2. I f J is a symmetric center set as described in Theorem 4.1, the corresponding graph I" is a regular graph with { ker J I (~) vertices (the size of J ) and valence 2(m - - 2) J ker J ]. Notice that
4(number of vertices)/valence = m + 1 + 2/(m -- 2).
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Thus with the restriction that m ~> 3 and m @ 4, m @ 1 is the integral part of this expression except when it is already integral, in which case m = 3. Consequently m and I ker J I (which is a power of 2) are determined by the parameters of F. T o produce/~ in the abstract, start with the graph whose vertices are the pairs (two-element subsets) of an m-set, two pairs being joined by an edge exactly when they share a member. T h e n replace each vertex by [ ker J I disconnected vertices, with vertices in different replacements being adjacent if and only if the vertices they replaced were adjacent. (This is the composition of the pair graph and a totally disconnected graph on ]ker J I vertices, in the sense of Harary [8, p. 22].) T h e duplication in types mentioned earlier runs as follows: m ~ 3: symplectic, dim V' = 2. m -- 5: quadratic, dim V' = 4, index =- 1. m ~- 6: symplectic, dim V' = 4. m -- 8: quadratic, dim V' -- 6, index = 3. These coincidences are classical; they are discussed by Dickson and Dieudonn6 [4, 6, 7]. This potpourri of conclusions from the list will be used later: PROPOSITION4.3. LetJbe a nontrivial connected center set that is not of symmetric type. Then if h and j are distinct members of J not connected by an edge, the number of members of J adjacent to both h and j is at least 12. In addition, there is a member k of J adjacent to neither h nor j for which h -~-j @ h isin J. Finally, the space g spanned by J contains two orthogonal planes each of which has all its nonzero members in J.
Proof. T h e restrictions from the list imply that dim V' ~ 6, since the other quadratic case when dim V' = 4 does not occur. Suppose J is of symplectic type. If h' = j ' in V', the number of v' for which ~b'(h', v') = ~b'(j', v') = 1 is F V' I/2. If h' =/=j', this number is [ V'//4. So in either case the number is at least 28/4, or 16. In V the number of v in J for which ~b(h, v) = ~b(j, v) = 1 is this quotient count times ] ker J ] and thus bigger than 12. For the quadratic type when h' = j', V' contains a subspace of the form (h', u') _[_ E 1 ]_ E 2 with all terms elliptic planes, because of the restriction on the index of V' when dim If' =- 6. The count for the needed v' lying in this subspace is 20. If h' ~ j', V' contains a subspace (h', u') ]_ ( j ' , w') _1_ E l , again with all terms elliptic planes. This time the count is 12. This case-by-case discussion also makes clear the existence of an element h in J for which h + j + h qlrad ~b. T h e n by Lemma 3.1, h + j + h ~ J . As for the third result, when Theorem 3.3 is used to describe J and the subspace V 0 of Section 2 is singled out, d i m V 0 = d i m V ' and so d i m V 0 >~6. T h e n V 0 contains two orthogonal elliptic planes and they fulfill the requirements.
CENTER SETS AND TERNARY CODES
217
5. CENTER SETS FROM TERNARY CODES
A (linear) code of length n over the finite field GF(q), q a prime power, is a subspace of the space GF(q)n of n-tuples with entries in GF(q). The n-tuples are called words of length n, and the members of a code under consideration, codewords. The weight of a word is the number of nonzero entries in it. T h e standard dot or inner product on GF(q) ~, as in Section 4, is the sum of component products. For a code C, its orthogonal code C ± is the subspace defined by
C± = { y ~ G F ( q ) ' ~ [ x . y = 0 f o r a l l x i n C } . C is called self-orthogonal if C C C l and self-dual (somewhat inaccurately) if C ~ C A. T h e support of a word is customarily taken as the subset of coordinate indices for which the corresponding entry in the word is not 0. But here it is more convenient to regard it as the binary word of the same length with 0's where the word has 0's and l's where it does not ("binary" means the entries are in GF(2)). T h e support of the word x i~ this sense will be written I x ]. Let xy stand for the component-wise product of the two words x and y, and let wt(x) denote the weight of x. In the binary case there is the easily-proved but useful formula
wt(x + y) = wt(x) -+- wt( y) -- 2wt(xy). (If x and y belong to the even subspace E of GF(2) ~ used in Section 4, then when this equation is divided by 2 and read mod 2, the result is
Q(x + y) -= Q(x) + Q( y) + x "y with Q(z) being wt(z)/2 read rood 2. In other words, half the weight function, mod 2, is a quadratic form on E whose symplectic form is the form~ of Section 4. This fact can be used to establish the two quadratic coincidences at the end of that section.) Now let C be a self-orthogonal ternary (q -= 3) linear code for which the m i n i m u m weight (of nonzero codewords) is 6. T h e hexads of C are the supports of the codewords of weight 6. PROPOSITION 5.1. Let C be a ternary self-orthogonal code with minimum weight 6, and let H be the set of hexads of C. Each member of H comes from exactly two members of C, and H is a center set relative to the standard binary inner product as the symplectic form.
Proof. If x and y are two words, they are said to "overlap" in the coordinate positions where neither is 0. If they agree in a of these positions and disagree in d, then a + d = wt(xy). When x and y are in C, the orthogonality, x • y = 0, requires that a ~ d (mod 3).
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Suppose now x and y have weight 6. If [ x [ = [ y ] but x v/= ~ y , then a = d ~ 3 and x + y has weight 3, which is too small. T h u s each hexad does come from exactly two words, negatives of one another. Examination of similar restrictions on a and d reveals that the overlap pattern of x and y (when x @ ~ y ) can only be one of the following: (1)
1 1 1 1 1 1 111
~ ) 1 1 1 1 1 1 12 (3) 0)
111
111
1 1 1 1 1 1 111
111
1 1 1 1 1 1 1 1 2 2
11
In these diagrams the words have been scaled and the coordinates permuted and scaled so that, among other things, x is represented by the first row. The sum ] x ] + lY r has weight 6 only in the thfrd arrangement, which is also the only one for w h i c h l x l - r y [ = 1. A n d i n t h a t c a s e l x l + l y l i s l x - - y [ . T h u s H is indeed a center set. THEOREM 5.2. Let K be a nontrivial connected component of the center set of hexads coming from a ternary self-orthogonal code C of minimum weight 6. Then K is of symmetric type.
Proof. The restriction on K is a euphemism for saying that K is the vertex set of a connected component of the graph of the hexad center set. Suppose K is not of symmetric type, so that the conclusions in Proposition 4.3 hold for K. Let h and j be two distinct hexads of K that are orthogonal (not connected by an edge). T h e n corresponding words of weight 6 in C cannot be arranged as in (4) above. For if they could, consider words of weight 6 whose hexads are adjacent to both. There are essentially two possible types, listed as the third rows in the following displays: 1 1 1 1 1 1 1 1 2 2 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 11 1
11 1
1
In the first configuration, row 1 + row 2 -- row 3 has weight 3 and that is out. I n the second, the first two l's of the third word can be either under the l's
219
CENTER SETS AND TERNARY CODES
of the second word, as shown, or under the 2's. The third 1 can be in one of two positions, and the fourth nonzero digit (which may have to be a 2) also in one of two positions. Thus there are eight possible arrangements for the locations of the first four nonzero digits of the third word. Since there must be at least 12 connecting hexads, two must come from words of weig'at 6 with the same arrangement. But proper scaling then yields two words of weight 6 agreeing in at least four positions of overlap without being scalar multiples. That goes against the four possibilities in the proof of Proposition 5.1. Therefore two orthogonal hexads of K overlap in two or no positions. Now the three term extension of the binary weight formula is
wt(x + y + z) wt(x) + wt(y) + wt(z) -- 2(wt(xy) + wt(xz) + wt(yz)) + 4wt(xyz). Let h and j be orthogonal hexads of K and k another that is orthogonal to both and for which h -k j -k k ~ K, as promised by Proposition 4.3. Then
wt(hj) + wt(hk) + wt(jk) -~ 6 + 2wt(hjk). As each term on the left is 0 or 2, they must all be 2 and wt(hjk) == 0. That is, two orthogonal hexads of K must overlap in two positions. Consider one of the elliptic planes at the end of Proposition 4.3. By scaling and permuting coordinates, words of weight 6 whose hexads are its three nonzero members can be arranged to look like the first three rows below: 1 1
1 1
1 1
1
1
1
1
1 2
1
1 1 1 222 2
1 1
1
1
Any word of weight 6 representing a hexad of K orthogonal to the elliptic plane will have to be arranged like the last row (with obvious positional and scaling choices), because of the needed overlaps of two. This means that any hexad orthogonal to the plane has three 1's outside the first nine positions. But the sum of two such hexads loses that property, and the second elliptic plane cannot be set up. Thus the assumption that K is not symmetric leads to an incompatibility. 6.
AN EXTREMAL CODE
The self-dual linear ternary code of length 16 and minimum weight 6 [3, Section VI] provides an illustration of the edge of Theorem 5.2, in a sense to be explained. One may say "the" code because up to permuting and scaling coordinates there is only one. Gleason's theorem [9, p. 620; 10] gives the weight
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HAROLD N. WARD
enumerator of such a code and it depends only on the length, 16, and the weight, 6. That enumerator predicts 224 words of weight 6. Building this code in the light of the preceding results requires determining the valence of the corresponding hexad graph; the graph happens to be regular. The proof of that parallels the proof of the Assmus-Mattson theorem [1, p. 139]. Namely, if C is the code and S is the support of a word of weight 6 (this time as a set of coordinate positions), the code C s of length 10 obtained from C by leaving off the coordinates in S is the orthogonal code of C °@s, the code of length 10 obtained from the words in C that are 0 at the positions in S by again leaving off the coordinates in S. C s has no words of weight 1, since such a word could only come from one of weight 6 in C having five nonzero entries in S. C °®s has dimension 3 and the weights of its words are 0, 6, or 9. The weight distributions of these two shortened codes are now forced by the MacWilliams relations [9, Chapter 5]. C s comes out to have 24 words of weight 3. An overlap study like that in the proof of Proposition 5.1 reveals that such a word comes from three in C: one of weight 9 and two of weight 6, as in this diagram: 1 1
1 1
222
1 1
1
1
i
222 1 1
1
1 1 1
1 1 1
1 1 1
The top row represents the word supported on S. Thus with three l's on S is also 24, so that the hexad graph valence 24. Now Corollary 4.2 provides the possible sizes of center sets wit~ valence 24. T h e y are sorted according [kerJ[ = m= IJl=91
1
2
4
14
8
5
56
40
the number of hexads is indeed regular with connected symmetric to I k e r J ] as follows:
Since the total number of hexads is 224/2, or 112, it can only be that there are two components, each with 56 members. This is why the code represents the "edge" of Theorem 5.2: if m 8, the center set is also quadratic. In general a nonzero member of the kernel of a center set of hexads, being the sum of two orthogonal hexads, will have weight 4, 8, or 12. Moreover, each hexad must overlap it in half of its positions in order that its sum with the hexad be another hexad. The beginning of the proof of Theorem 5.2 works in the present situation to show that two hexads adding to a member of the kernel cannot overlap in four places, because there are 24 hexads adjacent to both of them. Thus the nonzero member t 1 of the kernel for one component K 1 has weight 8 or 12. But if t 1 had weight 12, all 56 hexads of K 1 would be supported on its
CENTER SETS AND TERNARY CODES
221
12 positions. That would mean the subcode of C of words that are 0 outside those 12 positions would have at least 112 members--yet its dimension is only 4. Consequently t i has weight 8. Permute the coordinate positions so that the last eight constitute the support of t 1 . In the binary projection of/k~ onto the first eight positions, each hexad projects to a word of weight 2. Furthermore, the projections of adjacent hexads must overlap in exactly one position since their sum is also the projection of a hexad. According to the realizations of Theorem 4.1, K t contains seven mutually adjacent hexads h i ,..., h 7 , corresponding to the pair sums ei -~- e2 .... , ei q- e s (possibility (iii) does not apply since ker K1 @ 0). Their projections are words of weight 2, each two sharing one nonzero digit. It is then not hard to see that these projections must in fact all share a common position and make up the whole set of seven words of weight 2 having a 1 in that position. Permuting the first eight positions allows the assumption that the projection of ha has its nonzero entries in positions 1 and a. Take a ternary word corresponding to h,, and one corresponding to h~ q- t i . These two overlap in positions 1 and a, agreeing in one place and disagreeing in the other (for orthogonality). Thus their sum and difference are words of weight 9 with eight nonzero digits in the last eight positions and the ninth in position 1 in one case, position a in the other. That means that for each position among the first eight there is a word of weight 9 in C with a 1 there and its other nonzero digits in the last eight places. The matrix whose rows are eight such words, one for each of the first eight places and in proper order, is [I] HI, where I is the 8 by 8 identity and H is an 8 by 8 matrix with no O's. As the rank of the whole matrix is 8, it produces a basis for C. Because the sum and difference of two different rows both have weight at least 6, sum and difference have weight exactly 6. Thus two different rows of H disagree in four columns and agree in four. That is, H is an 8 by 8 Hadamard matrix (except for the fact that its entries are in GF(3); they should be read as l's and --l's). As everyone knows or can puzzle out, there is only one such matrix, apart from row and column scaling and permuting [16, p. 418]. Thus the code under consideration in this section must be unique--if it exists! But if H is an 8 by 8 Hadamard matrix, read into GF(3), the code spanned by the rows of [I1 H] has dimension 8. It is self-orthogonal because H T H = - - I (in GF(3)). The fact that [ - - H r [I] also spans it shows the minimum weight to be 6, from the same argument used by Pless [12, Section III].
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HAROLD N. WARD
7. A RELATED BINARY CODE
The second component K 2 of the center set of hexads for the code C in Section 6 corresponds to the generator matrix [ - - H r l I] in the same way K 1 corresponds to [I] H]. Let V be the span of K 1 and K 2 as a symplectic space, with the form ¢ being the restriction of the standard inner product on GF(2) 16. (K1) and (K2) are orthogonal subspaces of V and the quotient V/rad ~b is the orthogonal sum of their images. Each image is isometric to E*, the 6-dimensional nonsingular space coming from the even subspace E of GF(2) 8 in accordance with Theorem 4.1. In fact the discussion of Section 6 effectively produced a map of (K1) onto E by projection on the first eight positions; projection on the second eight works for (K2). It follows that V/rad ~b has dimension 12. Let the accent ' denote a map of V onto the orthogonal sum {(x*, y*) ] x, y in E} of two copies of E*, with kernel rad ~b, arranged so that K~ = {(x*, 0) I x ~ P} and K~ = {(0, y*) [Y ~ P}, P the set of pair sums. For concreteness this map may be considered as arising via the projections mentioned. The kernel tad ¢ contains t 1 and the analogous word 12 for K 2 whose eight l's occupy the first eight positions. In the even subspace of GF(2) 16, V _C (t 1 , t2) ±, forcing dim Vto be at most 14. Thus in fact dim V = 14 and rad ~b = ( t l , t2). The binary code B consists of all words of length 32 of the form (v, x, y ) in which v ~ V, x e E, y e E , and v' = (x*,y*), If wl and w2 are in B, where w i = (vi , x~ , Yi), then
w,. w2 = ¢(vl, v2) + 4(xl, x2) + ¢(y~, y~)
= ¢'(vl, v~) + ¢*(xf, x*) + ¢*(yf, yg) =0.
Therefore B is self-orthogonal. Projection of B on the second 16 coordinate positions has a 14-dimensional image and a 2-dimensional kernel (the space {(v, 0, 0) I v ~ rad ¢}). Thus B has dimension 16 and it is self-dual. B is spanned by the words of the form (kl, p, 0) and (k2,0, p), where k~ ~ K~ and k~ = p*; p 6 P. All of these have weight 8, so that the words in B all have weights divisible by 4 [9, p. 27]. Moreover, 8 is the minimum weight of B. For if (v, x, y) were in B and had weight 4, then since each component has even weight, one of them would be 0. But then v', x* and y* would all be 0 and each of v, x, and y would actually have weight divisible by 8. B is consequently an extremal doubly-even self-dual binary code of length 32. There are five such codes, up to equivalence, and they are described by Conway and Pless [2]. Two of them are well-known, the extended quadratic residue code q~2 (in their notation) and the second-order Reed-Muller code r~ [9, Chapters 15 and 16]. B turns out to be the code labeled 2g16, and here is a sketchy demonstration.
CENTER SETS AND TERNARY CODES
223
L e t F and G be the two 8 by 8 binary matrices displayed: 0 1 1 1 F = 1 1 1 1
0 1
1 1 1 1 G = 1 1 1 1
1
0
0
0
0
0
0
1 1 1 1 1 1 1 1
1 1 1
1
1 1
1 1
1 1 1
1 1 1 1
1
1 1
1 1
1
F is the identity matrix plus a first column of l's and G is the Paley-Hadamard matrix of order 8 [9, p. 48] with the -l's changed to O's. The description given for B produces the following generator matrix in block form:
o] F
0
The matrices F, G, and G r are bordered circulants, so that B is invariant under a permutation of coordinates of order 7 leaving the initial positions of each group of eight coordinate positions fixed. B must thus be among the codes r3~, 2gin, and 8f4, the only ones having automorphisms of order 7. There are 620 words of weight 8 in each of these codes, and by the Assmus-Mattson theorem, any three positions are in the supports of seven words of weight 8. With this as a guide one finds that the four fixed positions are not in a word of weight 8 of B. But they are for r32, from its Reed-Muller description; the words of weight 8 are the characteristic functions of the 3-dimensional flats in the 5-dimensional GF(2)-space on which the code is based. And they are for 8f4 , from the generator matrix given by Conway and Hess and its relation to their code 8d4 . Consequently B must be equivalent to the third code, 2916.
REFERENCES
1. E. F. ASSMUS,JR. ANDH. F. MATTSON,Jm, New 5-designs, J. Comb~atorial Theory 6 (1969), 122-151. 481/65/x-I5
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2. J. H. CONWAY AND V. PLESS, On the enumeration of self-dual codes, J. Combinatorial Theory 28 (1980), 26-53. 3. J. H. CONWAY, V. PLESS, AND N. J. A. SLOAN'E, Self-dual codes over GF(3) and GF(4) of length not exceeding 16, IEEE Trans. Information Theory IT-25 (1979), 312-322. 4. L. E. DICKSON, "Linear Groups, with an Exposition of the Galois Field Theory," Teubner, Leipzig, 1901 (reprinted Dover, New York, 1958). 5. L. E. DICKSON, Representations of the general symmetric group as linear groups in finite and infinite fields, Trans. Amer. Math. Soc. 9 (1908), 121-148. 6. J. DIEUDONNI~, "La g6om6trie des groupes classiques," 2nd ed., Ergebnisse der Math. No. 5, Springer-Verlag, Berlin, 1963. 7. J. DIEUDONNI~, Sur les groupes classiques, Actualitds Sci. Indust. 1040 (1948). 8. F. HARAR¥, " G r a p h Theory," Addison-Wesley, Reading, Mass., 1969. 9. F. J. MAcWILLIAMS AND N. J. A. SLOANE, " T h e Theory of Error-Correcting Codes," North-Holland, Amsterdam, 1977. 10. C. L. MALLOWS AND N. J. A. SLOANE, An upper bound for self-dual codes, Inform. Contr. 22 (1973), 188-200. 11. J. McLAUGHLIN, Some subgroups of SLn (F2), Illinois J. Math. 13 (1969), 108-115. 12. V. PLESS, Symmetry codes over GF(3) and new five-designs, ]. Combinatorial Theory 12 (1972), 119-142. 13. V. PLESS, N. J. A. SLOANE,AND H. N. WARD, Ternary codes of m i n i m u m weight 6 and the classification of the self-dual codes of length 20, IEEE Trans. Information Theory IT-26 (1980), 305-316. 14. R. RASALA, Split codes and the Matheiu groups, J. Algebra 42 (1976), 422-471. 15. N. J. A. SLOANE,Self-dual codes and lattices, in "Relations Between Combinatorics and Other Parts of Mathematics," pp. 273-308, Proc. Symp. Pure Math. Vol. 34, Amer. Math. Soc., Providence, R.I., 1979. 16. W. D. WALLIS, A. P. STREET, AND J. 8. WALLIS, "Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices," Lecture Notes in Mathematics No. 292, Springer-Verlag, Berlin, 1972. 17. H. N. WARD, Binary views of ternary codes, hz " T h e Geometric Vein: Essays Presented to H. S. M. Coxeter," to appear.
(1980)
Center Sets and Ternary Codes HAROLD N. WARD* Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903 Communicated by Walter Felt Received September 3, 1979 TO H . W .
BRINKMANN, ON THE OCCASION OF
HIS EIGHTIETH BIRTHDAY
INTRODUCTION In recent years the classification of self-dual codes has been carried almost to the limits on lengths beyond which the number of codes becomes overwhelming. The survey article by Sloane [15] refers to most of the papers making up this project; and the work by Pless, Sloane, and me [13], for which the present paper provides supporting material, brings his bibliography up to date. The approach used in that joint paper, which deals with codes over GF(3) or "ternary" codes, is typical of the classification for other fields. Let me give it a brief sketch. Decomposable codes of a given length are by definition direct sums of shorter codes and so determined by previous classifications. If an indecomposable code contains a fair number of words of weight 3 (all the words in a self-dual ternary code have weights divisible by 3), it will contain a large known decomposable subcodc. Additional codewords then serve to "glue" the components of this subcode together. The entire code can be constructed by a systematic study of possible glue words. (There is a good exposition of the glueing strategy in the paper of Conway, Pless, and Sloane [3].) When there are few or no words of weight 3, finding suitable decomposable subcodes can be difficult. One tries to construct the codes themselves by any means, fair or foul. At the same time the group of each code in hand is worked out so that the number of codes equivalent to it is known. Because the entire number of codes sought can be predicted, one knows when to stop. In fact, the nature of the difference when one is still short of the total can serve as a clue to the automorphism structure of missing codes (this is nicely illustrated by Conway and Pless [2]). Consider now a self-orthogonal ternary code having minimum weight 6, the * This research was partially supported by National Science Foundation Grant MCS 78-01458. 206 0021-8693/801070206-19502.00/0 Copyright © 1980 by Academic Press, Inc.
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largest possible minimum weight for self-dual codes of lengths at most 20. Each word of weight 6 determines a "hexad", the binary word obtained by changing any - - l ' s to l's and pretending the result has GF(2) entries. These hexads have the property that the sum, over GF(2), is again a hexad if and only if they share three l's; or, what turns out the same, if and only if their dot product is 1. They form an example of what is called a "center set". The combinatorial aspects of center sets restrict the possible disposition of the hexads enough to facilitate significantly the construction of codes. This technique was employed at the "fair or foul" step of our paper. Section 1 of the present paper contains the definition and basic properties of center sets. Each center set involves a graph in a natural way, and the first goal of the paper is the description of center sets with connected graphs. A review of properties of binary quadratic spaces and the construction of one class of center sets occupies Section 2. McLaughlin's classification of irreducible groups generated by transvections [11] provides the tool used in Sections 3 and 4 to classify connected center sets. Section 5 links ternary self-orthogonal codes and center sets. The major result is that the center sets arising are of symmetric type; that is, they involve the symmetric group in McLaughlin's classification. With this section the background for the application to self-dual codes of length 20 in our paper is complete. In that paper the reader will find examples of code constructions making use of the detailed descriptions of center sets to arrive at generating matrices. In Section 6 of this paper, however, I have given a construction of the unique self-dual ternary code of length 16 and minimum weight 6 to illustrate the kind of insight the center set approach provides. Part of the point is to give an example of a connection between ternary and binary codes that is a consequence of the center set perspective. The combinatorial structure of the hexads for this particular code leads to a binary self-dual code of length 32 and minimum weight 8, and the details are in the last section of the paper. The code constructed is different from two well-known examples, the Reed-Muller code and the extended quadratic residue code. Another instance of this kind of connection is that between the ternary extended Golay code of length 12 and the binary extended Golay code of length 24. One view of this was given by Rasala [14]; the view using center sets is set out in my Coxeter Symposium paper [17]. For background material in coding theory the reader should consult the book by MacWilliams and Sloane [9], to which several explicit references are made later on. In this paper the dimension of a vector space V is denoted dim V, and the subspace spanned by a set X is ( X ) . Displays of row vectors or matrices have blank spaces for zeroes except perhaps for selected positions used to clarify the pattern.
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HAROLD N. WARD 1. CENTER SETS
Center sets are certain subsets of symplectic vector spaces of finite dimension over the binary field GF(2). Let V be such a space and let ~bbe the corresponding symplectic form. It may be that ~b is degenerate; its radical is given b y rad~ ={vcV
I~(u,v) =0foralluin
V}.
(Standard references for symplectic spaces as well as for the material on quadratic spaces needed in Section 2 are the books of Dickson and Dieudonn4 [4, 6, 7].) DEFINITION |.1. A center set is a nonempty subset J of a symplectic space over GF(2) with this property: whenever u and v belong to J, u q- v is in J if and only if ~b(u, v) ~ 1. T h e members of J can be taken as the vertices of an undirected graph F in which two vertices u and v are joined by an edge exactly when ~b(u, v) - - 1. T h u s if u and v are adjacent (connected by an edge), u, v, and u ~ v will form a triangle. J also produces a linear group L on the vector space V containing J , generated by the transvections
tu: v --~ v + ~(v, U)U for each u in J - rad ~b (the "transvections" obtained from the u in rad ~b are each the identity). PROPOSITION 1.2. Let J be a center set and L its group. Then L preserves J, and the L-orbits of J are exactly the vertex sets of the connected components of the graph F of J. Each component has diameter at most 2.
Proof. If u ~ J - rad ~b, v ~ jr, and t~(v) ~ v, then ~b(u, v) - - 1 and tu(V ) = u ~ v, a m e m b e r of jr. L consequently preserves J ; and since ~b(u @ v, v) - - 1, u @ v is in the connected component containing v. T h u s the L-orbit of v is in that component. On the other hand, if w is adjacent to v, v @ w is in J and w = tv+w(V). It follows that each component is a n L - o r b i t . F o r the diameter statement, note that if u, x, y, and v are successively adjacent members of J for which u and y are not adjacent and x and v are not adjacent, then x @ y is in J and is adjacent to u and v. So any path joining two members of J can be modified step-by-step until it has length at most 2. W h e n J is a connected center set (one whose graph is connected) the sets {t~rad¢ I v @ t~J}, determined by each v in J , will not depend on v because of the transitivity of L on J and the fact that L acts trivially on rad ~b. Consequently the c o m m o n set T is a subspace of tad ~b. F o r if t 1 and t 2 are in T and v ~ jr, then v -t- tl ~ J and
CENTER SETS AND TERNARY CODES
209
then (v + tl) + t 2 e J also. That is, v + (q + t2) e J and t 1 + t 2 ~ T. Of course 0 E T; incidentally, 0 is never in a center set. DEFINITION 1.3. Let J be a connected center set. The kernel of J is the subspace ker J, where kerJ={t~rad¢[v+t~JforeachvinJ}. It will appear later that when J spans V, ker J is most of rad ¢. PROPOSITION 1.4. Let J be a connected center set spanning the corresponding space V. I f t: v ~ . v + ¢(~, u)u
is a transvection in the group L of J, then the center u of t belongs to J. Proof. By convention u 6 rad ¢, so that t(v) ~ v for some v in J, as J spans V. T h e n t ( v ) = u + v and u + v ~ J. But then u = ( u + v ) + v a n d u c J . I f J i n the proposition is not trivial, that is, not just a singleton set, J c~ rad ¢ = ~ , since something in rad ¢ would not be adjacent to anything else. The proposition says that J is exactly the set of centers of the transvections of L. This is the reason for using the name "center set". Let V' -- V/rad ¢, with the accent also denoting the quotient mapping. The form ¢' induced by ¢ on V' is nonsingular, as long as ¢ is not identically 0. L acts on V', producing theilinear group L' generated by the transvections with centers in J ' . PROPOSITION 1.5. I f J is a connected center set spanning V, then L' acts irreducibly on V'. Moreover, J ' is the set of centers of the transvections in L'. Proof. An L'-invariant subspace of V' is the image W' of an L-invariant subspace W of V containing rad ¢. If w c W - - rad ¢, then for some v in J, ¢(% w) = 1. Then v + w = t~(w), so that v - / w is in W, too. Thus v ~ W, and by the transitivity of L on J, all of J i s in W a n d W = V. Now J ' may not be a center set, but at least it has the property that if u' and v' are in J ' and ¢'(u', v') 1, then u' + v' 6 J ' . T h e second statement of the proposition follows just as in the proof of Proposition 1.4.
2. QUADRATIC SPACES Proposition 1.5 points to the use of McLaughlin's description of irreducible groups generated by transvections [11] in classifying connected center sets. Preparatory to that, let me summarize some properties of quadratic spaces over GF(2).
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On a finite-dimensional vector space V over GF(2), a quadratic form whose associated symplectic form is ~b is a function Q: V ~ GF(2) for which
Q(u + v) = Q(u) + Q(v) + 4J(u, v) for all u and v in V. A subspace and the quadratic form induced on it by restriction of Q are called singular if ~b is degenerate there, and totally singular if Q vanishes on the subspace. A nonsingular plane (two-dimensional subspace) may have just one vector on which Q is 1, in which case the plane is hyperbolic; if not, Q will have the value 1 on all three nonzero vectors and the plane is elliptic. Any quadratic space (one endowed with a quadratic form) that is nonsingular is an orthogonal sum of elliptic and hyperbolic planes. It is completely determined up to isometry by its dimension 2n and its index, the common dimension of its maximal totally singular subspaces. That index is n or n -- I according as the number of elliptic planes, in an orthogonal decomposition into planes, is even or odd. If the quadratic space V is singular, choose a supplement V0 to rad ~b, so that V is the orthogonal sum Vo ± rad ~b. T h e n the quadratic form Q is given by
Q(o + r) = Qo(~) + A(r) for v in Vo and r in rad ~b. Here Qo is the restriction of Q to Vo and nonsingular, while A is the restriction to tad ~b and actually a linear functional. When A = 0, Qo can legitimately be thought of as the form induced by Q on V/rad ~b; but when A 4 = 0, proper choice of Vo allows Qo to be taken with either of the two possible indices (provided Vo =/= 0). An important theorem is the one of Witt (which has an analogue for symplectic spaces): if V is a nonsingular quadratic space, any isometry between two subspaces can be extended to one of V. T h e appeals to Witt's theorem in the sequel involve constructing from a given spanning set of V a subspace isometric to the particular one being dealt with at the time. That the needed constructions can be made will usually be taken as evident. T h e results that follow illustrate the preceding ideas; Vo and A will continue to be used as above. PROPOSITION 2.1. Let V be a finite-dimensional quadratic space over GF(2), with quadratic form Q and associated symplectic form ~b. Then if the set J is nonempty, where
j = {v ~ V]Q(v) = 1 and v • rad ~b), it is a center set. Proof. Saying that J is nonempty is the same as saying rad ~b 4: V, that is, ~bis not identically 0. If u and v are in J, Q(u + v) -~ ~b(u, v). Thus if u + v E J,
CENTER SETS AND TERNARY CODES
211
~b(u, v) = 1. A n d if ~b(u, v) = 1, u + v e J provided u + v q~rad ~b. But since ~b(u, u + v) = ~b(u, v) =/= 0, this is so. PROPOSITION 2.2. Suppose that J in Proposition 2.1 is nonempty. Then ( J ) - V unless V o can be taken as a hyperbolic plane. Proof. If V o is the hyperbolic plane (u, v), with Q(u) and Q(v) both 0, then J is the union of the sets u + (rad ~b - - her A), v + (rad ~b - - her A), and u + v ~- her A (the first two may be empty). T h u s J does not span V. On the other hand suppose V 0 cannot be arranged as a hyperbolic plane, and let u ¢ rad ~b and u ¢ J . T h e n Q(u) = 0, and one may choose V o to have u in it. By W i t t ' s theorem there is a hyperbolic plane (u, v ) in V 0 . Because (u, v ) =/= Vo, there is a m e m b e r w of V o for which Q(w) 1 and w c (u, v ) ± (if V o is written as an orthogonal sum of planes one of which is (u, v), w may be selected from one of the other terms). T h e n w ~ J and u + w e J, so that u e ( J ) . ( j ) therefore contains all of V - - tad ~b; as that is more than half of V, ( J ) = V. Evidently J would still be a center set if the excluded members of rad ~b were admitted. T h e trouble is that any such elements will form isolated components in the graph. T h e next proposition settles connectivity for the original J. PROPOSITION 2.3. I f J is a nonempty center set of the kind in Proposition 2.1, then J is connected unless h = 0 and either V o is a hyperbolic plane and rad ~b 4 : 0 or else V o has dimension 4 and index 2. Proof. First suppose A @ 0. Let ./'1 and J2 be two distinct members of J, where j~ = vi + r i , vi E V0, and ri ~ rad ~b. In the verification that Jl and J2 are in the same connected component it may be assumed that ~b(j1 ,J2) ~ 0, that is, that ~b(vl, v~) = 0. If v 1 = v2, take v in V 0 for which ~b(v~, v) = 1 ; and if v~ ~ v2, use W i t t ' s theorem to produce a v in V o for which ~b(Vl, v) = ~b(vz, v) = 1. (In the second instance @1, v~) is isotropic and 2-dimensional.) T h e n take j to be v 4- r with r chosen in rad ~b to put j in J ; Q(r) = 1 + Q(v) and of course v 4- 0. T h e equations 4~(jl , j ) = 4J(j2 , j ) = 1 show Jl and J2 to be in the same component. N o w let A = 0. I f V o is a hyperbolic plane (u, v ) as in Proposition 2.2, J is the coset u + v + rad ~b and is totally disconnected when not a singleton set. I f V 0 = E1 5_ E2, the orthogonal sum of two elliptic planes, J is the union of the six cosets v + t a d ~b, where v =/= 0 and either v E E1 or v e E~. T h i s time J breaks into two components. I n any but these cases the argument is as before: again letj~ and j2 be distinct members of J , where ~b(j~ ,J2) = 0 and j i = vi + r , . I f V 1 = ' U 2 , select j in Vo making @1 , J ) an elliptic plane (so Q ( j ) = 1). I f v 1 4 v2, take u 1 and u 2 in V o making @1, u l ) elliptic, @2, u2) hyperbolic, and the two planes orthogonal. T h e n let j be u 1 + u 2 . T h e troublesome possibilities for Q0 for the application
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of Witt's theorem are those discussed above. In either situation, j ~ J ~ ( J l , J ) == ~(J2,J) -- 1, establishing the result. T h e proof of the last item of this section is routine: PROPOSITION 2.4.
and
For the center set of Proposition 2.1, the kernel is ker A.
3.
CLASSIFYING
CENTER
SETS
Let J be a connected center set spanning its vector space V. Assume that J is not trivially a singleton set, so that V 4: rad 4. By Proposition 1.5 the group L ' acts irreducibly on V'. L' is generated by the transvections with centers in J ' and J ' is the set of centers of the transvections in L'. T h e work of McLaughlin [11] yields these possibilities: (1) Symplectic type: L' is the symplectic group of the form ¢' induced by ¢ on V', and J ' is the set of all nonzero members of V'. (2) Quadratic t y p e : L ' is the orthogonal group of a quadratic form O' on V' having 4' as its associated symplectic form. T h e dimension of V' is at least 4, but the case that dim V' = 4 and O' has index 2 is missing. J' is the set of members of V' on which O' has value 1. T h e possibility that dim V' = 6 and Q' has index 3 will also be ruled out (and listed with the next type) for reasons that will become apparent. (3) Symmetric type: L' is isomorphic to the symmetric group on at least five letters. T h e details of this type will be given in Section 4. T h e types in this classification will be ascribed to the center sets themselves. T h e center sets of Section 2 account for the symplectic and quadratic types, principally because of the next result. LEMMA 3.1. Let J be a nontrivial connected center set spanning its space V, so that V :/= rad ¢. Suppose J is of symplectic or quadratic type. If j l , J2 , and j~ are three mutually orthogonal members of J and Jl + J2 + J3 ~ rad 4, then in fact
Jl + J2 + J~ e J. .t
.t
.t
Proof. In V', Jl + J~ + J~ =# 0. If the Ji are dependent, they might all coincide. In that case take j in J to satisfy ¢'(j~, j ' ) = 1, with (j~, j ' ) an elliptic plane when J is of quadratic type. If thej~ are dependent but not all coincident, •t .t ! t it must be that on renumbering j'~ 7~ J2 but j~ = j~. T h e n take u 1 and u2 so that ! / .t .t .t (J1, ul) and \J2, u~) are orthogonal nonsingular planes; (J1,32) is an isotropic t • .t t plane. This time if J is quadratic make (j~, ut) elllptm and (J2, u2) hyperbolic. T h e only block to doing that is the situation that dim V' = 4 and Q' has index 2, the first excluded possibility in the quadratic case. Then u'1 + u'2 c J ' because Q'(u~) = 1 and Q'(u~) = 0. L e t j be a preimage of u~ + u'2 in J. .t
-i
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AND
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213
If the j'£ are independent, construct three mutually orthogonal nonsingular .t t planes ( J i , ui), each one elliptic in the quadratic case; that involves the second exclusion. Again u'1 + u'2 @ u'a ~ J ' and j is taken in J satisfying j ' = u'1 +
u'2 + u'3.
For all possibilities the j produced is in J and ~(j, ji) = I for i = 1, 2, 3. Then, successively, j, j + j l , J+Jl+J2, J+JI+L+J3, and finally j~ @ jz + ja are all in J. Notice that if J is quadratic, j~ + j2 + J ~ ~ rad ~b automatically, because Q'(jl + J~ + J'3) = 1. And the conclusion of the last paragraph is of some use by itself: COROLLARY 3.2. t° Jl , J2 , and ja .
With the assumptions of Lemma 3.1, there is a j in J adjacent
THEOREM 3.3. Let J be a connected center set spanning its space V, where V @ rad ~b. Then i f J i s ofsymplectic or quadratic type, it is one of the sets described in Proposition 2.1. The quadratic form producing J is given by Q(v) = 1 if v ~ j or v e rad ~b -- ker J, O(v) = 0 otherwise. Proof. As the formula suggests, the proof amounts to showing that Q is really a quadratic form having ~b as its associated symplectic form. Because J spans V, it is enough to verify that Q(v + j) = Q(v) + Q(j) + ~b(v,j) for all v in V and j in J. If v a J the equation is Q(v + j) = ~b(v,j) and its correctness is almost just the definition of a center set. However, one needs to observe that if ~(v, j) = 0, then v + j is in ker J if it is in rad ~b, since its sum with j is back in J. When v E rad ~b, the equation becomes Q(v + j) = Q(v) -? 1 and it is a recasting of Definition 3.1 for ker J. Suppose then v is neither in J nor in rad ~b. Write v as a sum of members of J using as few as possible. If two terms were not orthogonal their sum would be in J and the number of terms could be reduced; so the terms are orthogonal. T h e n if there were three or more terms, L e m m a 3.1 would imply that the sum of any three of them is in tad ~b, otherwise the sum could again be shortened. As v q~rad ~b, there could not be exactly three terms; but if there were more than three, the sums of pairs (as sums of overlapping groups of three) would also be in rad q~. A little thought reveals that v would end up in rad ~b. Consequently v is the sum jl +/'2 of two orthogonal members of J. If ~b(v,j) ~ 1, then ~b(jl,j) = 1 and ~b(j2,j) = 0 , say. Thus v + j =
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(/'1 + J) + / ' 2 , a sum of two orthogonal members of J. So v ~- j is in ker J if it is in tad ¢. Hence Q(v -k j ) = Q(v) = o and the equation is correct. If ¢(% j) = 0 and both ¢(Jl ,J) and ¢(/'2 ,J) are 0, then as v j - j ----/'1 -kj~ + j , L e m m a 3.1 implies that v + j is either in J or tad ¢. But if it is in rad ¢, it is not in ker J, because it plus j is not in J. This time, then, Q(v) = 0 and Q(v -[- j) = 1 as needed. Finally, if ¢(/'1, J) -- ¢(J~, J) - 1, Jl + J and then/'1 +/'2 -k j are in J and the equation is again correct.
4. SYMMETRIC CENTER SETS For a center set of symmetric type, the g r o u p L ' is isomorphic to the symmetric group on m letters for a certain m. I n the division of types in Section 3, m > / 5 . It is profitable to relax that restriction, even though there is some duplication with the other types (which will be discussed later). Henceforth a symmetric center set will mean one in which L' is isomorphic to a symmetric group acting on V' in the manner about to be described, with the corresponding J ' . Let S be the symmetric group on m letters, the "letters" being 1, 2,..., m. Let GF(2) m be the space of m-tuples over GF(2), with standard basis e1 ..... era; e~ is the m-tuple with a 1 in the i-th place and O's elsewhere. T h e action of S on GF(2) m is given by
g(al ..... am) = (ag-~(1) ,..., ag-l(m)) when g ff Sin. Thus g(ei) = eg(i) , S preserves the standard inner product on GF(2) m which is the sum of the component products. It also preserves the even subspace E of m-tuples having component sum 0, On E the inner product is a symplectic form ¢, and the radical of ¢ is 0 if m is odd, but the span of the all-1 m-tuple if m is even. Let * stand for the map of E onto the quotient E* of E by the radical of ¢, and let ¢* be the induced form on E*; it is nondegenerate as long as m ) 3. For a symmetric center set, V' is isometric to E* endowed with the form ¢*, for an appropriate m. Furthermore, if P is the set of pair sums ea -? eb, where a v~ b, J ' under the isometry is P*. The representation of S involved here was studied by Dickson [5]. This description of V' leads in turn to one of V and J ; the discussion that follows will be summed up in Theorem 4.1. When m is odd, identify V' with E .t and J ' with P. Take Jla in J so that Jla = el + ca, where a > 1. T h e n when •! .t a > I, b > 1, and a @ b , it follows that ¢(Jla,Jlb) = ¢ ( J l a , J l ~ ) = 1 and jl~ -? Jib is in J. Put
L~ =Ao +Ab. From this, j~b -----ea -}- eb, and if Vo is the span of the jab, Vo will be isometric
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to E and V = V o 5_ rad ~b. T h e members of J are the members of the cosets
jab + ker J , and since J is supposed to span V, ker J = rad ~b. T h u s when m is odd, V can be taken as E _[_ R, the form ¢ being 6 on E and 0 on R. T h e kernel of J is R and J is the union of the cosets p + R, where p E P. Notice that the transvection t , on E coming from a pair sum p in P is the transformation produced by the transposition (ab) in S, where p --~ ea + e~. N o w suppose m is even. Let e ~ e1 + ' " + e m be the all-1 m-tuple, so that r a d ¢ = (e). V' is to be identified with E* and J ' with P*; m ~ 4. Again select Jla in J for which J~a ~- e* + e* and let ja~ = ]la + jlb" I f ker J = o, v is spanned by the jab and the kernel of the h o m o m o r p h i s m of V onto E* is (J12 + "'" + jim). I f this is 0, V can be taken as E*, and if not, as E. J is P* or P in the two cases. I f ker J is not 0, there is some flexibility in the choices of thejaa's, and one can make sure that if w : Jl~ + "'" + J l - , , w :/: 0. T h e n if V 0 is the span of the jab , Vo is isometric to E. Because J is the union of the cosets jab + ker J , v = v o+kerJ and V ~ V 0 _ l _ R for some subspaee R of k e r J . I n addition, rad ¢ : ( w ) _1_ R. T h e sum w may or m a y not be in ker J . W h e t h e r ker J is 0 or not, when m : 6 the sum J12 + J13 -]- "'" + Jl~ cannot be in k e r J ; in particular it is not 0. F o r that sum is j l ~ +J84 +J56; were it in ker J , its sum j8 a -+-J56 with J12 would be in J . But ¢(J34, J56) -/: 1. W h e n m : 3 or m : 4 the center set is also of symplectic type, with dim V' : 2. T h u s one can also assume m 4- 4 (the set-up for 3 is easier to visualize!) T h e final s u m m a r y of symmetric center sets is this: THEOREM 4.1. Let J be a nontrivial connected center set of symmetric type, spanning its vector space V. Then up to isometry V and J are one of the following arrangements, with the notation as above. R is a finite-dimensional GF(2)-space carrying the 0 form. (i)
V=E_l_R,J={p+rlpeP,
(ii) V = E ± R , J = { p + r , p + e + r l p e P , and m is even and at least 8. (iii)
rER};kerJ=R,
andm>/3, m~4.
r~R};kerJ=(e)+R,
If = E*, J = P*; ker J = 0 and again m is even and at least 8.
Verifying that these descriptions do give connected center sets of symmetric type is straight-forward. F o r applications the parameters of the corresponding graphs need to be given. COROLLARY 4.2. I f J is a symmetric center set as described in Theorem 4.1, the corresponding graph I" is a regular graph with { ker J I (~) vertices (the size of J ) and valence 2(m - - 2) J ker J ]. Notice that
4(number of vertices)/valence = m + 1 + 2/(m -- 2).
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HAROLD N. WARD
Thus with the restriction that m ~> 3 and m @ 4, m @ 1 is the integral part of this expression except when it is already integral, in which case m = 3. Consequently m and I ker J I (which is a power of 2) are determined by the parameters of F. T o produce/~ in the abstract, start with the graph whose vertices are the pairs (two-element subsets) of an m-set, two pairs being joined by an edge exactly when they share a member. T h e n replace each vertex by [ ker J I disconnected vertices, with vertices in different replacements being adjacent if and only if the vertices they replaced were adjacent. (This is the composition of the pair graph and a totally disconnected graph on ]ker J I vertices, in the sense of Harary [8, p. 22].) T h e duplication in types mentioned earlier runs as follows: m ~ 3: symplectic, dim V' = 2. m -- 5: quadratic, dim V' = 4, index =- 1. m ~- 6: symplectic, dim V' = 4. m -- 8: quadratic, dim V' -- 6, index = 3. These coincidences are classical; they are discussed by Dickson and Dieudonn6 [4, 6, 7]. This potpourri of conclusions from the list will be used later: PROPOSITION4.3. LetJbe a nontrivial connected center set that is not of symmetric type. Then if h and j are distinct members of J not connected by an edge, the number of members of J adjacent to both h and j is at least 12. In addition, there is a member k of J adjacent to neither h nor j for which h -~-j @ h isin J. Finally, the space g spanned by J contains two orthogonal planes each of which has all its nonzero members in J.
Proof. T h e restrictions from the list imply that dim V' ~ 6, since the other quadratic case when dim V' = 4 does not occur. Suppose J is of symplectic type. If h' = j ' in V', the number of v' for which ~b'(h', v') = ~b'(j', v') = 1 is F V' I/2. If h' =/=j', this number is [ V'//4. So in either case the number is at least 28/4, or 16. In V the number of v in J for which ~b(h, v) = ~b(j, v) = 1 is this quotient count times ] ker J ] and thus bigger than 12. For the quadratic type when h' = j', V' contains a subspace of the form (h', u') _[_ E 1 ]_ E 2 with all terms elliptic planes, because of the restriction on the index of V' when dim If' =- 6. The count for the needed v' lying in this subspace is 20. If h' ~ j', V' contains a subspace (h', u') ]_ ( j ' , w') _1_ E l , again with all terms elliptic planes. This time the count is 12. This case-by-case discussion also makes clear the existence of an element h in J for which h + j + h qlrad ~b. T h e n by Lemma 3.1, h + j + h ~ J . As for the third result, when Theorem 3.3 is used to describe J and the subspace V 0 of Section 2 is singled out, d i m V 0 = d i m V ' and so d i m V 0 >~6. T h e n V 0 contains two orthogonal elliptic planes and they fulfill the requirements.
CENTER SETS AND TERNARY CODES
217
5. CENTER SETS FROM TERNARY CODES
A (linear) code of length n over the finite field GF(q), q a prime power, is a subspace of the space GF(q)n of n-tuples with entries in GF(q). The n-tuples are called words of length n, and the members of a code under consideration, codewords. The weight of a word is the number of nonzero entries in it. T h e standard dot or inner product on GF(q) ~, as in Section 4, is the sum of component products. For a code C, its orthogonal code C ± is the subspace defined by
C± = { y ~ G F ( q ) ' ~ [ x . y = 0 f o r a l l x i n C } . C is called self-orthogonal if C C C l and self-dual (somewhat inaccurately) if C ~ C A. T h e support of a word is customarily taken as the subset of coordinate indices for which the corresponding entry in the word is not 0. But here it is more convenient to regard it as the binary word of the same length with 0's where the word has 0's and l's where it does not ("binary" means the entries are in GF(2)). T h e support of the word x i~ this sense will be written I x ]. Let xy stand for the component-wise product of the two words x and y, and let wt(x) denote the weight of x. In the binary case there is the easily-proved but useful formula
wt(x + y) = wt(x) -+- wt( y) -- 2wt(xy). (If x and y belong to the even subspace E of GF(2) ~ used in Section 4, then when this equation is divided by 2 and read mod 2, the result is
Q(x + y) -= Q(x) + Q( y) + x "y with Q(z) being wt(z)/2 read rood 2. In other words, half the weight function, mod 2, is a quadratic form on E whose symplectic form is the form~ of Section 4. This fact can be used to establish the two quadratic coincidences at the end of that section.) Now let C be a self-orthogonal ternary (q -= 3) linear code for which the m i n i m u m weight (of nonzero codewords) is 6. T h e hexads of C are the supports of the codewords of weight 6. PROPOSITION 5.1. Let C be a ternary self-orthogonal code with minimum weight 6, and let H be the set of hexads of C. Each member of H comes from exactly two members of C, and H is a center set relative to the standard binary inner product as the symplectic form.
Proof. If x and y are two words, they are said to "overlap" in the coordinate positions where neither is 0. If they agree in a of these positions and disagree in d, then a + d = wt(xy). When x and y are in C, the orthogonality, x • y = 0, requires that a ~ d (mod 3).
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HAROLD N. WARD
Suppose now x and y have weight 6. If [ x [ = [ y ] but x v/= ~ y , then a = d ~ 3 and x + y has weight 3, which is too small. T h u s each hexad does come from exactly two words, negatives of one another. Examination of similar restrictions on a and d reveals that the overlap pattern of x and y (when x @ ~ y ) can only be one of the following: (1)
1 1 1 1 1 1 111
~ ) 1 1 1 1 1 1 12 (3) 0)
111
111
1 1 1 1 1 1 111
111
1 1 1 1 1 1 1 1 2 2
11
In these diagrams the words have been scaled and the coordinates permuted and scaled so that, among other things, x is represented by the first row. The sum ] x ] + lY r has weight 6 only in the thfrd arrangement, which is also the only one for w h i c h l x l - r y [ = 1. A n d i n t h a t c a s e l x l + l y l i s l x - - y [ . T h u s H is indeed a center set. THEOREM 5.2. Let K be a nontrivial connected component of the center set of hexads coming from a ternary self-orthogonal code C of minimum weight 6. Then K is of symmetric type.
Proof. The restriction on K is a euphemism for saying that K is the vertex set of a connected component of the graph of the hexad center set. Suppose K is not of symmetric type, so that the conclusions in Proposition 4.3 hold for K. Let h and j be two distinct hexads of K that are orthogonal (not connected by an edge). T h e n corresponding words of weight 6 in C cannot be arranged as in (4) above. For if they could, consider words of weight 6 whose hexads are adjacent to both. There are essentially two possible types, listed as the third rows in the following displays: 1 1 1 1 1 1 1 1 2 2 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 11 1
11 1
1
In the first configuration, row 1 + row 2 -- row 3 has weight 3 and that is out. I n the second, the first two l's of the third word can be either under the l's
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CENTER SETS AND TERNARY CODES
of the second word, as shown, or under the 2's. The third 1 can be in one of two positions, and the fourth nonzero digit (which may have to be a 2) also in one of two positions. Thus there are eight possible arrangements for the locations of the first four nonzero digits of the third word. Since there must be at least 12 connecting hexads, two must come from words of weig'at 6 with the same arrangement. But proper scaling then yields two words of weight 6 agreeing in at least four positions of overlap without being scalar multiples. That goes against the four possibilities in the proof of Proposition 5.1. Therefore two orthogonal hexads of K overlap in two or no positions. Now the three term extension of the binary weight formula is
wt(x + y + z) wt(x) + wt(y) + wt(z) -- 2(wt(xy) + wt(xz) + wt(yz)) + 4wt(xyz). Let h and j be orthogonal hexads of K and k another that is orthogonal to both and for which h -k j -k k ~ K, as promised by Proposition 4.3. Then
wt(hj) + wt(hk) + wt(jk) -~ 6 + 2wt(hjk). As each term on the left is 0 or 2, they must all be 2 and wt(hjk) == 0. That is, two orthogonal hexads of K must overlap in two positions. Consider one of the elliptic planes at the end of Proposition 4.3. By scaling and permuting coordinates, words of weight 6 whose hexads are its three nonzero members can be arranged to look like the first three rows below: 1 1
1 1
1 1
1
1
1
1
1 2
1
1 1 1 222 2
1 1
1
1
Any word of weight 6 representing a hexad of K orthogonal to the elliptic plane will have to be arranged like the last row (with obvious positional and scaling choices), because of the needed overlaps of two. This means that any hexad orthogonal to the plane has three 1's outside the first nine positions. But the sum of two such hexads loses that property, and the second elliptic plane cannot be set up. Thus the assumption that K is not symmetric leads to an incompatibility. 6.
AN EXTREMAL CODE
The self-dual linear ternary code of length 16 and minimum weight 6 [3, Section VI] provides an illustration of the edge of Theorem 5.2, in a sense to be explained. One may say "the" code because up to permuting and scaling coordinates there is only one. Gleason's theorem [9, p. 620; 10] gives the weight
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enumerator of such a code and it depends only on the length, 16, and the weight, 6. That enumerator predicts 224 words of weight 6. Building this code in the light of the preceding results requires determining the valence of the corresponding hexad graph; the graph happens to be regular. The proof of that parallels the proof of the Assmus-Mattson theorem [1, p. 139]. Namely, if C is the code and S is the support of a word of weight 6 (this time as a set of coordinate positions), the code C s of length 10 obtained from C by leaving off the coordinates in S is the orthogonal code of C °@s, the code of length 10 obtained from the words in C that are 0 at the positions in S by again leaving off the coordinates in S. C s has no words of weight 1, since such a word could only come from one of weight 6 in C having five nonzero entries in S. C °®s has dimension 3 and the weights of its words are 0, 6, or 9. The weight distributions of these two shortened codes are now forced by the MacWilliams relations [9, Chapter 5]. C s comes out to have 24 words of weight 3. An overlap study like that in the proof of Proposition 5.1 reveals that such a word comes from three in C: one of weight 9 and two of weight 6, as in this diagram: 1 1
1 1
222
1 1
1
1
i
222 1 1
1
1 1 1
1 1 1
1 1 1
The top row represents the word supported on S. Thus with three l's on S is also 24, so that the hexad graph valence 24. Now Corollary 4.2 provides the possible sizes of center sets wit~ valence 24. T h e y are sorted according [kerJ[ = m= IJl=91
1
2
4
14
8
5
56
40
the number of hexads is indeed regular with connected symmetric to I k e r J ] as follows:
Since the total number of hexads is 224/2, or 112, it can only be that there are two components, each with 56 members. This is why the code represents the "edge" of Theorem 5.2: if m 8, the center set is also quadratic. In general a nonzero member of the kernel of a center set of hexads, being the sum of two orthogonal hexads, will have weight 4, 8, or 12. Moreover, each hexad must overlap it in half of its positions in order that its sum with the hexad be another hexad. The beginning of the proof of Theorem 5.2 works in the present situation to show that two hexads adding to a member of the kernel cannot overlap in four places, because there are 24 hexads adjacent to both of them. Thus the nonzero member t 1 of the kernel for one component K 1 has weight 8 or 12. But if t 1 had weight 12, all 56 hexads of K 1 would be supported on its
CENTER SETS AND TERNARY CODES
221
12 positions. That would mean the subcode of C of words that are 0 outside those 12 positions would have at least 112 members--yet its dimension is only 4. Consequently t i has weight 8. Permute the coordinate positions so that the last eight constitute the support of t 1 . In the binary projection of/k~ onto the first eight positions, each hexad projects to a word of weight 2. Furthermore, the projections of adjacent hexads must overlap in exactly one position since their sum is also the projection of a hexad. According to the realizations of Theorem 4.1, K t contains seven mutually adjacent hexads h i ,..., h 7 , corresponding to the pair sums ei -~- e2 .... , ei q- e s (possibility (iii) does not apply since ker K1 @ 0). Their projections are words of weight 2, each two sharing one nonzero digit. It is then not hard to see that these projections must in fact all share a common position and make up the whole set of seven words of weight 2 having a 1 in that position. Permuting the first eight positions allows the assumption that the projection of ha has its nonzero entries in positions 1 and a. Take a ternary word corresponding to h,, and one corresponding to h~ q- t i . These two overlap in positions 1 and a, agreeing in one place and disagreeing in the other (for orthogonality). Thus their sum and difference are words of weight 9 with eight nonzero digits in the last eight positions and the ninth in position 1 in one case, position a in the other. That means that for each position among the first eight there is a word of weight 9 in C with a 1 there and its other nonzero digits in the last eight places. The matrix whose rows are eight such words, one for each of the first eight places and in proper order, is [I] HI, where I is the 8 by 8 identity and H is an 8 by 8 matrix with no O's. As the rank of the whole matrix is 8, it produces a basis for C. Because the sum and difference of two different rows both have weight at least 6, sum and difference have weight exactly 6. Thus two different rows of H disagree in four columns and agree in four. That is, H is an 8 by 8 Hadamard matrix (except for the fact that its entries are in GF(3); they should be read as l's and --l's). As everyone knows or can puzzle out, there is only one such matrix, apart from row and column scaling and permuting [16, p. 418]. Thus the code under consideration in this section must be unique--if it exists! But if H is an 8 by 8 Hadamard matrix, read into GF(3), the code spanned by the rows of [I1 H] has dimension 8. It is self-orthogonal because H T H = - - I (in GF(3)). The fact that [ - - H r [I] also spans it shows the minimum weight to be 6, from the same argument used by Pless [12, Section III].
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HAROLD N. WARD
7. A RELATED BINARY CODE
The second component K 2 of the center set of hexads for the code C in Section 6 corresponds to the generator matrix [ - - H r l I] in the same way K 1 corresponds to [I] H]. Let V be the span of K 1 and K 2 as a symplectic space, with the form ¢ being the restriction of the standard inner product on GF(2) 16. (K1) and (K2) are orthogonal subspaces of V and the quotient V/rad ~b is the orthogonal sum of their images. Each image is isometric to E*, the 6-dimensional nonsingular space coming from the even subspace E of GF(2) 8 in accordance with Theorem 4.1. In fact the discussion of Section 6 effectively produced a map of (K1) onto E by projection on the first eight positions; projection on the second eight works for (K2). It follows that V/rad ~b has dimension 12. Let the accent ' denote a map of V onto the orthogonal sum {(x*, y*) ] x, y in E} of two copies of E*, with kernel rad ~b, arranged so that K~ = {(x*, 0) I x ~ P} and K~ = {(0, y*) [Y ~ P}, P the set of pair sums. For concreteness this map may be considered as arising via the projections mentioned. The kernel tad ¢ contains t 1 and the analogous word 12 for K 2 whose eight l's occupy the first eight positions. In the even subspace of GF(2) 16, V _C (t 1 , t2) ±, forcing dim Vto be at most 14. Thus in fact dim V = 14 and rad ~b = ( t l , t2). The binary code B consists of all words of length 32 of the form (v, x, y ) in which v ~ V, x e E, y e E , and v' = (x*,y*), If wl and w2 are in B, where w i = (vi , x~ , Yi), then
w,. w2 = ¢(vl, v2) + 4(xl, x2) + ¢(y~, y~)
= ¢'(vl, v~) + ¢*(xf, x*) + ¢*(yf, yg) =0.
Therefore B is self-orthogonal. Projection of B on the second 16 coordinate positions has a 14-dimensional image and a 2-dimensional kernel (the space {(v, 0, 0) I v ~ rad ¢}). Thus B has dimension 16 and it is self-dual. B is spanned by the words of the form (kl, p, 0) and (k2,0, p), where k~ ~ K~ and k~ = p*; p 6 P. All of these have weight 8, so that the words in B all have weights divisible by 4 [9, p. 27]. Moreover, 8 is the minimum weight of B. For if (v, x, y) were in B and had weight 4, then since each component has even weight, one of them would be 0. But then v', x* and y* would all be 0 and each of v, x, and y would actually have weight divisible by 8. B is consequently an extremal doubly-even self-dual binary code of length 32. There are five such codes, up to equivalence, and they are described by Conway and Pless [2]. Two of them are well-known, the extended quadratic residue code q~2 (in their notation) and the second-order Reed-Muller code r~ [9, Chapters 15 and 16]. B turns out to be the code labeled 2g16, and here is a sketchy demonstration.
CENTER SETS AND TERNARY CODES
223
L e t F and G be the two 8 by 8 binary matrices displayed: 0 1 1 1 F = 1 1 1 1
0 1
1 1 1 1 G = 1 1 1 1
1
0
0
0
0
0
0
1 1 1 1 1 1 1 1
1 1 1
1
1 1
1 1
1 1 1
1 1 1 1
1
1 1
1 1
1
F is the identity matrix plus a first column of l's and G is the Paley-Hadamard matrix of order 8 [9, p. 48] with the -l's changed to O's. The description given for B produces the following generator matrix in block form:
o] F
0
The matrices F, G, and G r are bordered circulants, so that B is invariant under a permutation of coordinates of order 7 leaving the initial positions of each group of eight coordinate positions fixed. B must thus be among the codes r3~, 2gin, and 8f4, the only ones having automorphisms of order 7. There are 620 words of weight 8 in each of these codes, and by the Assmus-Mattson theorem, any three positions are in the supports of seven words of weight 8. With this as a guide one finds that the four fixed positions are not in a word of weight 8 of B. But they are for r32, from its Reed-Muller description; the words of weight 8 are the characteristic functions of the 3-dimensional flats in the 5-dimensional GF(2)-space on which the code is based. And they are for 8f4 , from the generator matrix given by Conway and Hess and its relation to their code 8d4 . Consequently B must be equivalent to the third code, 2916.
REFERENCES
1. E. F. ASSMUS,JR. ANDH. F. MATTSON,Jm, New 5-designs, J. Comb~atorial Theory 6 (1969), 122-151. 481/65/x-I5
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2. J. H. CONWAY AND V. PLESS, On the enumeration of self-dual codes, J. Combinatorial Theory 28 (1980), 26-53. 3. J. H. CONWAY, V. PLESS, AND N. J. A. SLOAN'E, Self-dual codes over GF(3) and GF(4) of length not exceeding 16, IEEE Trans. Information Theory IT-25 (1979), 312-322. 4. L. E. DICKSON, "Linear Groups, with an Exposition of the Galois Field Theory," Teubner, Leipzig, 1901 (reprinted Dover, New York, 1958). 5. L. E. DICKSON, Representations of the general symmetric group as linear groups in finite and infinite fields, Trans. Amer. Math. Soc. 9 (1908), 121-148. 6. J. DIEUDONNI~, "La g6om6trie des groupes classiques," 2nd ed., Ergebnisse der Math. No. 5, Springer-Verlag, Berlin, 1963. 7. J. DIEUDONNI~, Sur les groupes classiques, Actualitds Sci. Indust. 1040 (1948). 8. F. HARAR¥, " G r a p h Theory," Addison-Wesley, Reading, Mass., 1969. 9. F. J. MAcWILLIAMS AND N. J. A. SLOANE, " T h e Theory of Error-Correcting Codes," North-Holland, Amsterdam, 1977. 10. C. L. MALLOWS AND N. J. A. SLOANE, An upper bound for self-dual codes, Inform. Contr. 22 (1973), 188-200. 11. J. McLAUGHLIN, Some subgroups of SLn (F2), Illinois J. Math. 13 (1969), 108-115. 12. V. PLESS, Symmetry codes over GF(3) and new five-designs, ]. Combinatorial Theory 12 (1972), 119-142. 13. V. PLESS, N. J. A. SLOANE,AND H. N. WARD, Ternary codes of m i n i m u m weight 6 and the classification of the self-dual codes of length 20, IEEE Trans. Information Theory IT-26 (1980), 305-316. 14. R. RASALA, Split codes and the Matheiu groups, J. Algebra 42 (1976), 422-471. 15. N. J. A. SLOANE,Self-dual codes and lattices, in "Relations Between Combinatorics and Other Parts of Mathematics," pp. 273-308, Proc. Symp. Pure Math. Vol. 34, Amer. Math. Soc., Providence, R.I., 1979. 16. W. D. WALLIS, A. P. STREET, AND J. 8. WALLIS, "Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices," Lecture Notes in Mathematics No. 292, Springer-Verlag, Berlin, 1972. 17. H. N. WARD, Binary views of ternary codes, hz " T h e Geometric Vein: Essays Presented to H. S. M. Coxeter," to appear.