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Automorphisms of hyperelliptic GAG-codes
Electronic Notes in Discrete Mathematics 26 (2006) 123–130 www.elsevier.com/locate/endm
Automorphisms of hyperelliptic GAG-codes Alberto Picone 1 and Antonino Giorgio Spera 2 Dipartimento di Matematica ed Applicazioni Universit` a degli Studi di Palermo Via Archirafi N◦ 34, 90123 Palermo, Italy
Abstract We determine the n−automorphism group of generalized algebraic-geometry codes associated with rational, elliptic and hyperelliptic function fields. Such group is, up to isomorphism, a subgroup of the automorphism group of the underlying function field. Keywords: Geometric Goppa codes, generalized algebraic-geometry codes, algebraic function fields, automorphisms, finite fields. Math. Sub. Cl. (2000): Primary 94B27, Secondary 14H45.
1
Introduction and preliminaries
Generalized algebraic-geometry codes (in short GAG−codes) have recently been introduced by Xing, Niederreiter and Lam [13] and they have allowed us to obtain codes with better parameters compared with Brouwer’s table (see [1]). They are a generalization of the well-known geometric Goppa codes since, for their construction, we can use not only rational places but also place of higher degree as well. For our purposes we are interested in GAG−codes 1 2
Email: [email protected] Email: [email protected]
1571-0653/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2006.08.022
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constructed with places which are all of the same degree. For such GAG−code, Spera [8] proved that its automorphism group admits a subgroup which is isomorphic to an automorphism group of the underlying function field. This result is similar to a result by Stichtenoth [10] on geometric Goppa codes. In the present paper, following Wesemeyer [11] (see also [4]) and introducing the notion of n−automorphism for GAG−codes, we are able to invert Spera’s afore-mentioned result in the rational, elliptic and hyperelliptic cases. We show that, under certain suitable conditions, the n−automorphism group of a GAG−code can be embedded in the automorphism group of the underlying function field. Now we recall some properties of GAG−codes. For all notations about algebraic function fields, geometric Goppa codes and coding theory, which are not explicitly defined, we use the standard definitions as in [9] and in [12]. Let F |Fq be a function field of genus g. A φ − place is a pair (P, φP ) where P is a place of F |Fq and φP : FP → Fqdeg P is a Fq −isomorphism (see [8]). It is defined an action of Aut(F |Fq ) on the set of the φ−places by σ((P, φP )) := (σ(P ), φP σ −1 ) where σ ∈ Aut(F |Fq ). A φ−divisor is an element of the free group generated by φ−places. Aut(F |Fq ) acts also on the group of the φ−divisors in obvious way. N Let Φ = i=1 (Pi , φi ) be a φ−divisor of F |Fq with Pi pairwise distinct places all of the same degree n > 1. Let G be a divisor such that Supp G ∩ nN {P1 , P2 , . . . PN } = ∅. Let’s consider the map evΦ : L(G) → FN such q n = Fq that for all z ∈ L(G), evΦ (z) := (z(P1 , φ1 ), z(P2 , φ2 ), . . . , z(PN , φN )) where z(Pi , φi ) := φi (z + Pi ). We will call the linear code C(Φ; G; n) := evΦ (L(G)) a generalized algebraicgeometry code (in short GAG-code). The following proposition is in [7]. Proposition 1.1 If deg G < nN then C(Φ; G; n) is a q−ary [nN, k, d] code with k = dim G ≥ deg G + 1 − g and d ≥ N − degn G (where x denotes the greatest integer smaller than or equal to x). Note that the GAG-codes defined above are a particular class of the more general GAG-codes considered in [13] and also in [3]. Let Φ = N i=1 (Pi , φi ) and let G and G be divisors of F |Fq . G is said to be Φ−equivalent to G (G ∼Φ G ) if and only if there exists an element z ∈ F , z = 0, such that G = G + (z) and z(Pi , φi ) = 1 for every i = 1, 2, . . . , N . Of course ∼Φ is an equivalence relation and we can define the subgroup Aut(F |Fq , Φ, G) = {σ ∈ Aut(F |Fq ) | σ(Φ) = Φ and σ(G) ∼Φ G} of Aut(F |Fq ). If G = G0 − G1 where G0 and G1 are effective divisors and if deg (G0 + G1 ) ≤
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nN −1 then Aut(F |Fq , Φ, G) is the stabilizer (see [8]) of Φ and G in Aut(F |Fq ). In [8] we also have the following result. Theorem 1.2 If C(Φ; G; n) is a GAG−code associated with Φ and G then: 1. Any automorphism σ ∈ Aut(F |Fq , Φ, G) induces an automorphism of C(Φ; G; n) by σ(z(P1 , φ1 ), . . . , z(PN , φN )) = (z(σ(P1 , φ1 )), . . . , z(σ(PN , φN ))) for every z ∈ L(G); 2. Aut(C(Φ; G; n)) has a subgroup which is isomorphic to Aut(F |Fq , Φ, G) 2 if N − 1 > 16(g+1) or F |Fq has some rational places and N ≥ 4g+3 . n n 3. If G = rQ, where Q is a rational place, r ≤ nN − 1 and 2g + 2 ≤ nN , then C(Φ; G; n) admits an automorphism group which is isomorphic to Aut(F |Fq )Φ,Q .
2
Automorphisms
Let n and m be positive integers and suppose n be a divisor of m. We can write every element of c ∈ Fm q like c = (a1 , a2 , . . . , am/n ) where ai = ci1 ci2 · · · cin with cij ∈ Fq for 1 ≤ i ≤ m/n and 1 ≤ j ≤ n. So Sm/n acts on Fm q via π(c) = (aπ(1) , aπ(2) , . . . , aπ(m/n) ) if π ∈ Sm/n and c = (a1 , a2 , . . . , am/n ) ∈ Fm q . What we are doing is to permute the components of c seen as m/n sequences of length n. Of course π is also an element of Sm . Moreover if n = 1 we get the usual action of Sm on Fm q . Let C(Φ; G; n) ⊆ FnN be a GAG-code associated with Φ and G. π ∈ S nN is q n said to be an n-automorphism of the code if π(c) = (aπ(1) , aπ(2) , . . . , aπ(N ) ) ∈ C(Φ; G; n) for any c = (a1 , a2 , . . . , aN ) ∈ C(Φ; G; n). The n-automorphism group H(Φ; G; n) of the GAG-code will be H(Φ; G; n) := {π ∈ SN | π(c) ∈ C(Φ; G; n) for any c ∈ C(Φ; G; n)}. Remark 2.1 H(Φ; G; n) is a subgroup of Aut(C(Φ; G; n)) and, by the point 1 of the Theorem 1.2, any element of Aut(F |Fq )Φ,G induces an n-automorphism. Moreover, points 2 and 3 of the same Theorem 1.2, give conditions so that Aut(F |Fq )Φ,G can be embedded into H(Φ; G; n). Lemma 2.2 Let u, v ∈ L(G) for some divisor G = G0 − G1 with G0 , G1 ≥ 0. Let r = deg G0 < nN . Let (Pi , φi ), for i = 1, 2, . . . , N , be some φ−places with Pi pairwise distinct places all of the same degree n, which are not in the support of G0 . If u(Pi , φi ) = v(Pi , φi ) for each i, then u = v. We are going to define now an Fq −liner map with which we will can associate to each n−automorphism of the code an element of Aut(F |Fq )Φ,G .
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N Let Φ = N i=1 (Pi , φi ) and Ψ = i=1 (Pi , ψi ) be two φ−divisors with deg Pi = deg Pi = n > 1 for any i = 1, 2, . . . , N . Let G be a divisor of degree deg G < nN such that Supp G ∩ {P1 , P2 , . . . , PN } = Supp G ∩ {P1 , P2 , . . . , PN } = ∅. Let’s consider the linear isomorphisms evΦ and evΨ . If C(Φ; G; n) = C(Ψ; G; n) we can define, as in [11], a linear isomorphism λ : L(G) → L(G) −1 by λ := evΨ ◦ evΦ . The next lemma shows that our map λ, under certain conditions on the degree of G, has similar properties with the ones of a field automorphism. Lemma 2.3 The following holds: 1. If 1 ∈ L(G) then λ(1) = 1; 2. Let G > 0 with deg G < nN . Then: 2 (a) if f, g ∈ L(G) are such that also f g ∈ L(G), then λ(f g) = λ(f )λ(g); (b) if f, f k ∈ L(G) for k ≥ 2 then λ(f k ) = λ(f )k and deg(λ(f )∞ ) ≤ degk G . The following proposition links λ to an automorphism of the function field. With it, we will show how λ can be used to associate to an automorphism of the code, an automorphism of the stabilizer. Proposition 2.4 Let F = Fq (x, y) be an algebraic function field and σ ∈ Aut(Fq (x, y)|Fq ). Let’s suppose that C(Φ; G; n) = C(Ψ; G; n) where Φ = N N (P , ψ ) and the support of G is disjoint from the i=1 (Pi , φi ), Ψ = N i=1 i i −1 supports of D := i=1 Pi and D := N i=1 Pi . As above let λ := evΨ ◦ evΦ and suppose x, y ∈ L(G). If σ(G) = G and σ|L(G) = λ then σ(Pi , φi ) = (Pi , ψi ) for each i = 1, 2, . . . , N . From now on we suppose that F |Fq is a hyperelliptic function field of genus g with char Fq = 2. We think elliptic function field as special case of hyperelliptic function field. For any reference see [9]. We have F = Fq (x, y) with H(x, y) = y 2 − f (x) = 0 and f (x) ∈ Fq [x] is a square-free polynomial of degree d = 2g + 1 or 2g + 2. The places P ∈ PFq (x) which ramify in F |Fq (x) are all zeros of f (x) if deg f (x) ≡ 0(mod2) and also the pole P∞ of x if deg f (x) ≡ 1(mod2). The other places stay inert or split completely. Let D∞ be the divisor which is the sum of the places of F |Fq lying over the pole of x. If P is a rational place of F for which x and y are regular, then xP := x(P ) and yP := y(P ) uniquely determine P . We identify P with (xP , yP ). The automorphism group of F |Fq (x) is equal to Aut(F |Fq (x)) = {id, ξ} where ξ is the nontrivial automorphism such that ξ(x) = x and ξ(y) = −y. For the conjugate P := ξ(P ) of P , we have P = (xP , −yP ).
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Let’s set H(Fq ) := PF ∪ supp(D∞ ), Hr (Fq ) the set of the places of H(Fq ) \ supp(D∞ ) which ramify in F |Fq and Hs (Fq ) the set of the places of H(Fq ) \ supp(D∞ ) which split. Finally we can prove the main theorem. Theorem 2.5 Let J ⊆ H(Fq ) \ supp ((x)∞ ) and let G = n∞ D∞ + Q∈J nQ Q ⎧ ⎨ 2g + 2 if d ≡ 1(mod2) be a divisor, where n∞ ≥ and nQ ≥ 1 for each ⎩ g + 1 if d ≡ 0(mod2) Q ∈ J. Let Φ = N i=1 (Pi , φi ) be a φ−divisor with Pi distinct places all of the same degree n > 1. Let deg G < nN . If nN > 16(g + 1)2 + n or F |Fq has 2 some rational place and nN ≥ 4g + 3, then H(Φ; G; n)) ∼ = Aut(F |Fq )Φ,G . Proof By the Theorem 1.2, Aut(Fq (x, y)|Fq )Φ,G is isomorphic to a subgroup of H(Φ; G; n). We are going to prove that such subgroup is actually isomorphic to the whole group H(Φ; G; n). So let’s consider π ∈ H(Φ; G; n). It is easy to show that C(Φ; G; n) = C(πΦ; G; n) and so we can consider the corresponding map λ = λπ . Now for m := g + 1, by Proposition 4.5 in [11], a base of L(G) is B = xα y β |α ≥ 0, β ∈ {0, 1} and xα y β ∈ L(n∞ D∞ ),
α
β y 1 |2α + β ≤ nQ , α ≥ 0, β ∈ {0, 1} and Q ∈ J ∩ Hr (Fq ), x−xQ x−xQ (gmQ )α (gβQ )|mα + β ≤ nQ , α ≥ 0, 0 ≤ β < m and Q ∈ J ∩ Hs (Fq ) y−hi (x) (x−xP )i
∈ F with deg hi (x) ≤ i − 1, vP (y − hi (x)) ≥ i, vP (y − ⎧ ⎨ −(2g + 1 − 2i), if d ≡ 1(mod2) hi (x)) = 0, vP (giP ) = −i and v∞ (giP ) ≥ ⎩ −(g + 1 − i), if d ≡ 0(mod2) (where v∞ (h) = Q∈supp (D∞ ) vQ (h)). α β So, by Lemma 2.3, it is possible that λ(x y ) = λ(x)α λ(y)β , for to prove
α
β
α
β λ(y) y 1 1 xα y β ∈ L(G). We also have λ = x−xQ x−xQ λ(x)−xQ λ(x)−xQ where giP :=
for 2α + β ≤ nQ , α ≥ 0, β ∈
α {0, 1} and Q ∈ J ∩ Hr (Fq ). Moreover λ(y)−pβ (λ(x)) m (λ(x)) λ((gmQ )α (gβQ )) = λ(y)−p for mα + β ≤ nQ , α ≥ 0, (λ(y)−xQ )i (λ(y)−xQ )i 0 ≤ β < m and Q ∈ J ∩ Hs (Fq ). Since every element h(x, y) ∈ L(G) can be written as a linear combination of the above elements, we proved that λ(h(x, y)) = h(λ(x), λ(y)). We will prove now that λ(x) and λ(y) satisfy the relation H(λ(x), λ(y)) = λ(y)2 − f (λ(x)) = d+1 0. We start noticing that x and x[ 2 ] belong to L(G) and so, by Lemma 2.3,
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t we have deg (λ(x))∞ ≤ g+1 ≤ 2td that is deg ((λ(x)d )∞ ) ≤ 2t < nN from which follows, since deg f (x) = d, that deg(f (λ(x)))∞ ≤ 2t. Moreover, since λ(y) ∈ L(G) we have deg (λ(y)2 )∞ = 2deg (λ(y))∞ ≤ 2t. So we have that λ(y)2 , f (λ(x)) ∈ L(2G). We also remark that λ(y)2 (P, φ) = f (λ(x))(P, φ) for each (P, φ) ∈ supp Φ, in fact: λ(y)2 (Pπ(i) , φπ(i) ) = φπ(i) (λ(y)2 + Pπ(i) ) = φπ(i) ((λ(y)+Pπ(i) )2 ) = (φπ(i) (λ(y)+Pπ(i) ))2 = (λ(y)(Pπ(i) , φπ(i) ))2 = (y(Pi , φi ))2 = y 2 (Pi , φi ) = f (x)(Pi , φi ) = φi (f (x) + Pi ) = φi (f (x + Pi )) = f (φi (x + Pi )) = f (x(Pi , φi )) = f (λ(x)(Pπ(i) , φπ(i) )) = f (φπ(i) (λ(x) + Pπ(i) )) = φπ(i) (f (λ(x) + Pπ(i) )) = φπ(i) (f (λ(x)) + Pπ(i) ) = f (λ(x))(Pπ(i) , φπ(i) ). Then, since deg 2G < nN , by Lemma 2.2, we have λ(y)2 = f (λ(x)). It is easy to show that the : Fq (x, y) → Fq (x, y) such that λ(x) = λ(x) and λ(y) = λ(y) is a map λ to L(G) Fq −endomorphism. It is obvious moreover that the restriction of λ is coincides with λ and so, since x and y are in the image of λ (x, y ∈ L(G)), λ ∈ Aut(F |Fq )Φ,G . But, an automorphism of F |Fq . We have only to prove that λ since G > 0 and λ(L(G)) = L(G), it follows that (see [11]) λ(G) = G and so it is possible to apply Lemma 2.4 from which follows that λ(Pi φi ) = (Pπ(i) , φπ(i) ) that is λ(Φ) = Φ. 2
The next theorem shows that similar result can be also obtained in the rational case. Theorem 2.6 Let Fq (x)|Fq be a rational function field. Let Φ = N i=1 (Pi , φi ) be a φ−divisorwith Pi pairwise distinct places all of the same degree n. Let G = rP∞ + m i=1 ri Pαi with r > 0, ri ≥ 0 and Pαi rational places. Let C(Φ; G; n) be the GAG−code associated with Φ and G. If deg G < nN and 2 nN ≥ 3 then H(Φ; G; n) ∼ Aut(F (x)|F ) . = q q Φ,G Now we give some constructions of rational GAG−codes and of their n−automorphism groups. Let Fq (x)|Fq be a rational function field. Let P = Pp(x) be a place of degree n 1 (where p(x) is a monic, irreducible polynomial of degree n). Let α ∈ Fq be a fixed root of p(x) (Fq denotes the algebraic closure of the finite field Fq ). 2 n−1 It is well-known that α, αq , αq , . . . , αq are exactly the n roots of p(x). So, one of its roots identifies clearly the polynomial p(x) and so, without creating ambiguity, it is possible to indicate Pp(x) with P[α] . With abuse of notation we will indicate also P∞ with P[∞] . Let FP = FPp(x) be the residue class field of P . There are exactly n Fq −isomorphisms between this field and the field Fqn . In fact, the map for all z + P = u(x) + P ∈ FP φα : FP → Fqn such that φα (z + P ) := u(α) v(α) v(x) is, clearly, a field Fq −isomorphism and the n Fq −isomorphisms between the
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(i)
fields FP and Fqn will be φα := ρi φα for i = 0, 1, . . . , n − 1, where ρ is the Frobenius’s automorphism of Fqn |Fq . (i)
qi
(i)
(i)
) , we have φα = φαqi and then (P[α] , φα ) = (P[αqi ] , φαqi ). Since φα (z+P ) = u(α v(αqi ) This means that, if we choose properly the root, we can always suppose a φ−place to be of the type (P[α] , φα ). It is well-known how the projective group P GL(2, q) acts on the projective line P G(1, q) = Fq ∪{∞}. Whereas its action on Fq (x)|Fq is defined as follows.
−1 f (σ[A] (x)) (x) (x) := g(σ−1 Let σ[A] ∈ P GL(2, q) and fg(x) ∈ Fq (x) then σ[A] fg(x) where (x)) [A] ⎛ ⎞ ab −1 dx−b ⎠ ∈ GL(2, q). σ[A] (x) = −cx+a if A = ⎝ cd It is well-known also that every projectivity acts like an Fq −automorphism of Fq (x) and that Aut(Fq (x)|Fq ) ∼ = P GL(2, q) (see for instance [6]). When σ[A] will denote an automorphism of Fq (x)|Fq it will be denoted with σA . Note that, since Fq ⊆ Fqn , we have that P GL(2, q) is, up to isomorphism, GL(2, q n ) ∼ a subgroup of P GL(2, q n ) (and so Aut(Fq (x)|Fq ) ∼ = P GL(2, q)⊆P = Aut(Fqn (x)|Fqn )). Thus P GL(2, q) acts on Fqn too. In fact, if σ[A] ∈ P GL(2, q) and α ∈ Fqn , we have that σ[A] (α) := aα+b if α ∈ / Fq , while σ[A] (α) is defined cα+d like usual if α ∈ Fq . Let’s remark that if α ∈ / Fq (and so n > 1) then cα+d = 0 and so σ[A] (α) is well defined. Finally, we can describe the action of Aut(Fq (x)|Fq ) on the set of φ−places of Fq (x)|Fq .
Proposition 2.7 If (P[α] , φα ) is a φ−place of Fq (x)|Fq and σA ∈ Aut(Fq (x)|Fq ) then σA (P[α] , φα ) = (P[σ[A] (α)] , φσ[A] (α) ). We will describe now the stabilizer Aut(Fq (x)|Fq )Φ,kP∞ and, with this, we can give a different version of the Theorem 2.6. Proposition 2.8 Let Φ = N i=1 (P[αi ] , φαi ) be a φ−divisor with P[αi ] distinct places all of degree n > 1 of the function field Fq (x)|Fq . Let G = rP∞ . Then the automorphisms σA ∈ Aut(Fq (x)|Fq ) which fix Φ and G are the affinities of the affine line which permute the roots αi . Corollary 2.9 Let Fq (x)|Fq be a rational function field and let C(Φ; G; n) be nN the GAG−code associated with Φ = N i=1 (P[αi ] , φαi ) and G = rP∞ . If r < 2 then H(n; Φ; G) is the group of the affinities of the affine line which permute the roots αi .
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References [1] A.E. Brouwer, Bounds on minimum distance of linar codes, [Online], Available: http://www.win.tue.nl/∼aeb/voorlincod.html. [2] V.D. Goppa, Codes on Algebraic Curves, Soviet Math. Dokl. 24, No. 1, 170-172 (1981). [3] A.E. Heydtmann, Generalized Geometric Goppa Codes, Comm. Algebra 30, No. 6 2763-2789 (2002). [4] D. Joyner and A. Ksir, Automorphism group of some AG Codes, available at http://arxiv.org/abs/AG/0412459, v2, (2005). [5] D. Jungnickel, Finite Fields-Structure and Arithmetics, BI-Wiss.-Verl., Maunheim, Leipzig, Wien, Z¨ urich 1993. [6] S. Roman, Field Theory, Springer-Verlag, New York-Berlin-Heidelberg 1995. [7] A.G. Spera, Asymptotically Good Codes from Generalized Algebraic-Geometric Codes, Designs, Codes and Cryptography, Vol. 37, No. 2, 305-312 (2005). [8] A.G. Spera, On Automorphisms of generalized Algebraic-Geometry Codes, To appear on Journal of pure and applied Algebra (2006). [9] H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, BerlinHeidelberg 1993. [10] H. Stichtenoth, On Automorphisms of Geometric Goppa Codes, J. of Algebra 130, 113-121 (1990). [11] S. Wesemeyer, On the Automorphism Group of Various Goppa Codes, IEEE Transactions on Information Theory, Vol. 44, No. 2, 630-643 (1998). [12] M.A. Tsfasman and S.G. Vladut, Algebraic-Geometric Codes, Kluwer Acad. Publ., Dordrecht-Boston-London 1991. [13] C.P. Xing, H. Niederreiter and K.Y. Lam, A Generalization of AlgebraicGeometric Codes, IEEE Transactions on Information Theory, Vol. 45, No. 7, 2438-2501 (1999).
Automorphisms of hyperelliptic GAG-codes Alberto Picone 1 and Antonino Giorgio Spera 2 Dipartimento di Matematica ed Applicazioni Universit` a degli Studi di Palermo Via Archirafi N◦ 34, 90123 Palermo, Italy
Abstract We determine the n−automorphism group of generalized algebraic-geometry codes associated with rational, elliptic and hyperelliptic function fields. Such group is, up to isomorphism, a subgroup of the automorphism group of the underlying function field. Keywords: Geometric Goppa codes, generalized algebraic-geometry codes, algebraic function fields, automorphisms, finite fields. Math. Sub. Cl. (2000): Primary 94B27, Secondary 14H45.
1
Introduction and preliminaries
Generalized algebraic-geometry codes (in short GAG−codes) have recently been introduced by Xing, Niederreiter and Lam [13] and they have allowed us to obtain codes with better parameters compared with Brouwer’s table (see [1]). They are a generalization of the well-known geometric Goppa codes since, for their construction, we can use not only rational places but also place of higher degree as well. For our purposes we are interested in GAG−codes 1 2
Email: [email protected] Email: [email protected]
1571-0653/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2006.08.022
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constructed with places which are all of the same degree. For such GAG−code, Spera [8] proved that its automorphism group admits a subgroup which is isomorphic to an automorphism group of the underlying function field. This result is similar to a result by Stichtenoth [10] on geometric Goppa codes. In the present paper, following Wesemeyer [11] (see also [4]) and introducing the notion of n−automorphism for GAG−codes, we are able to invert Spera’s afore-mentioned result in the rational, elliptic and hyperelliptic cases. We show that, under certain suitable conditions, the n−automorphism group of a GAG−code can be embedded in the automorphism group of the underlying function field. Now we recall some properties of GAG−codes. For all notations about algebraic function fields, geometric Goppa codes and coding theory, which are not explicitly defined, we use the standard definitions as in [9] and in [12]. Let F |Fq be a function field of genus g. A φ − place is a pair (P, φP ) where P is a place of F |Fq and φP : FP → Fqdeg P is a Fq −isomorphism (see [8]). It is defined an action of Aut(F |Fq ) on the set of the φ−places by σ((P, φP )) := (σ(P ), φP σ −1 ) where σ ∈ Aut(F |Fq ). A φ−divisor is an element of the free group generated by φ−places. Aut(F |Fq ) acts also on the group of the φ−divisors in obvious way. N Let Φ = i=1 (Pi , φi ) be a φ−divisor of F |Fq with Pi pairwise distinct places all of the same degree n > 1. Let G be a divisor such that Supp G ∩ nN {P1 , P2 , . . . PN } = ∅. Let’s consider the map evΦ : L(G) → FN such q n = Fq that for all z ∈ L(G), evΦ (z) := (z(P1 , φ1 ), z(P2 , φ2 ), . . . , z(PN , φN )) where z(Pi , φi ) := φi (z + Pi ). We will call the linear code C(Φ; G; n) := evΦ (L(G)) a generalized algebraicgeometry code (in short GAG-code). The following proposition is in [7]. Proposition 1.1 If deg G < nN then C(Φ; G; n) is a q−ary [nN, k, d] code with k = dim G ≥ deg G + 1 − g and d ≥ N − degn G (where x denotes the greatest integer smaller than or equal to x). Note that the GAG-codes defined above are a particular class of the more general GAG-codes considered in [13] and also in [3]. Let Φ = N i=1 (Pi , φi ) and let G and G be divisors of F |Fq . G is said to be Φ−equivalent to G (G ∼Φ G ) if and only if there exists an element z ∈ F , z = 0, such that G = G + (z) and z(Pi , φi ) = 1 for every i = 1, 2, . . . , N . Of course ∼Φ is an equivalence relation and we can define the subgroup Aut(F |Fq , Φ, G) = {σ ∈ Aut(F |Fq ) | σ(Φ) = Φ and σ(G) ∼Φ G} of Aut(F |Fq ). If G = G0 − G1 where G0 and G1 are effective divisors and if deg (G0 + G1 ) ≤
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nN −1 then Aut(F |Fq , Φ, G) is the stabilizer (see [8]) of Φ and G in Aut(F |Fq ). In [8] we also have the following result. Theorem 1.2 If C(Φ; G; n) is a GAG−code associated with Φ and G then: 1. Any automorphism σ ∈ Aut(F |Fq , Φ, G) induces an automorphism of C(Φ; G; n) by σ(z(P1 , φ1 ), . . . , z(PN , φN )) = (z(σ(P1 , φ1 )), . . . , z(σ(PN , φN ))) for every z ∈ L(G); 2. Aut(C(Φ; G; n)) has a subgroup which is isomorphic to Aut(F |Fq , Φ, G) 2 if N − 1 > 16(g+1) or F |Fq has some rational places and N ≥ 4g+3 . n n 3. If G = rQ, where Q is a rational place, r ≤ nN − 1 and 2g + 2 ≤ nN , then C(Φ; G; n) admits an automorphism group which is isomorphic to Aut(F |Fq )Φ,Q .
2
Automorphisms
Let n and m be positive integers and suppose n be a divisor of m. We can write every element of c ∈ Fm q like c = (a1 , a2 , . . . , am/n ) where ai = ci1 ci2 · · · cin with cij ∈ Fq for 1 ≤ i ≤ m/n and 1 ≤ j ≤ n. So Sm/n acts on Fm q via π(c) = (aπ(1) , aπ(2) , . . . , aπ(m/n) ) if π ∈ Sm/n and c = (a1 , a2 , . . . , am/n ) ∈ Fm q . What we are doing is to permute the components of c seen as m/n sequences of length n. Of course π is also an element of Sm . Moreover if n = 1 we get the usual action of Sm on Fm q . Let C(Φ; G; n) ⊆ FnN be a GAG-code associated with Φ and G. π ∈ S nN is q n said to be an n-automorphism of the code if π(c) = (aπ(1) , aπ(2) , . . . , aπ(N ) ) ∈ C(Φ; G; n) for any c = (a1 , a2 , . . . , aN ) ∈ C(Φ; G; n). The n-automorphism group H(Φ; G; n) of the GAG-code will be H(Φ; G; n) := {π ∈ SN | π(c) ∈ C(Φ; G; n) for any c ∈ C(Φ; G; n)}. Remark 2.1 H(Φ; G; n) is a subgroup of Aut(C(Φ; G; n)) and, by the point 1 of the Theorem 1.2, any element of Aut(F |Fq )Φ,G induces an n-automorphism. Moreover, points 2 and 3 of the same Theorem 1.2, give conditions so that Aut(F |Fq )Φ,G can be embedded into H(Φ; G; n). Lemma 2.2 Let u, v ∈ L(G) for some divisor G = G0 − G1 with G0 , G1 ≥ 0. Let r = deg G0 < nN . Let (Pi , φi ), for i = 1, 2, . . . , N , be some φ−places with Pi pairwise distinct places all of the same degree n, which are not in the support of G0 . If u(Pi , φi ) = v(Pi , φi ) for each i, then u = v. We are going to define now an Fq −liner map with which we will can associate to each n−automorphism of the code an element of Aut(F |Fq )Φ,G .
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N Let Φ = N i=1 (Pi , φi ) and Ψ = i=1 (Pi , ψi ) be two φ−divisors with deg Pi = deg Pi = n > 1 for any i = 1, 2, . . . , N . Let G be a divisor of degree deg G < nN such that Supp G ∩ {P1 , P2 , . . . , PN } = Supp G ∩ {P1 , P2 , . . . , PN } = ∅. Let’s consider the linear isomorphisms evΦ and evΨ . If C(Φ; G; n) = C(Ψ; G; n) we can define, as in [11], a linear isomorphism λ : L(G) → L(G) −1 by λ := evΨ ◦ evΦ . The next lemma shows that our map λ, under certain conditions on the degree of G, has similar properties with the ones of a field automorphism. Lemma 2.3 The following holds: 1. If 1 ∈ L(G) then λ(1) = 1; 2. Let G > 0 with deg G < nN . Then: 2 (a) if f, g ∈ L(G) are such that also f g ∈ L(G), then λ(f g) = λ(f )λ(g); (b) if f, f k ∈ L(G) for k ≥ 2 then λ(f k ) = λ(f )k and deg(λ(f )∞ ) ≤ degk G . The following proposition links λ to an automorphism of the function field. With it, we will show how λ can be used to associate to an automorphism of the code, an automorphism of the stabilizer. Proposition 2.4 Let F = Fq (x, y) be an algebraic function field and σ ∈ Aut(Fq (x, y)|Fq ). Let’s suppose that C(Φ; G; n) = C(Ψ; G; n) where Φ = N N (P , ψ ) and the support of G is disjoint from the i=1 (Pi , φi ), Ψ = N i=1 i i −1 supports of D := i=1 Pi and D := N i=1 Pi . As above let λ := evΨ ◦ evΦ and suppose x, y ∈ L(G). If σ(G) = G and σ|L(G) = λ then σ(Pi , φi ) = (Pi , ψi ) for each i = 1, 2, . . . , N . From now on we suppose that F |Fq is a hyperelliptic function field of genus g with char Fq = 2. We think elliptic function field as special case of hyperelliptic function field. For any reference see [9]. We have F = Fq (x, y) with H(x, y) = y 2 − f (x) = 0 and f (x) ∈ Fq [x] is a square-free polynomial of degree d = 2g + 1 or 2g + 2. The places P ∈ PFq (x) which ramify in F |Fq (x) are all zeros of f (x) if deg f (x) ≡ 0(mod2) and also the pole P∞ of x if deg f (x) ≡ 1(mod2). The other places stay inert or split completely. Let D∞ be the divisor which is the sum of the places of F |Fq lying over the pole of x. If P is a rational place of F for which x and y are regular, then xP := x(P ) and yP := y(P ) uniquely determine P . We identify P with (xP , yP ). The automorphism group of F |Fq (x) is equal to Aut(F |Fq (x)) = {id, ξ} where ξ is the nontrivial automorphism such that ξ(x) = x and ξ(y) = −y. For the conjugate P := ξ(P ) of P , we have P = (xP , −yP ).
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Let’s set H(Fq ) := PF ∪ supp(D∞ ), Hr (Fq ) the set of the places of H(Fq ) \ supp(D∞ ) which ramify in F |Fq and Hs (Fq ) the set of the places of H(Fq ) \ supp(D∞ ) which split. Finally we can prove the main theorem. Theorem 2.5 Let J ⊆ H(Fq ) \ supp ((x)∞ ) and let G = n∞ D∞ + Q∈J nQ Q ⎧ ⎨ 2g + 2 if d ≡ 1(mod2) be a divisor, where n∞ ≥ and nQ ≥ 1 for each ⎩ g + 1 if d ≡ 0(mod2) Q ∈ J. Let Φ = N i=1 (Pi , φi ) be a φ−divisor with Pi distinct places all of the same degree n > 1. Let deg G < nN . If nN > 16(g + 1)2 + n or F |Fq has 2 some rational place and nN ≥ 4g + 3, then H(Φ; G; n)) ∼ = Aut(F |Fq )Φ,G . Proof By the Theorem 1.2, Aut(Fq (x, y)|Fq )Φ,G is isomorphic to a subgroup of H(Φ; G; n). We are going to prove that such subgroup is actually isomorphic to the whole group H(Φ; G; n). So let’s consider π ∈ H(Φ; G; n). It is easy to show that C(Φ; G; n) = C(πΦ; G; n) and so we can consider the corresponding map λ = λπ . Now for m := g + 1, by Proposition 4.5 in [11], a base of L(G) is B = xα y β |α ≥ 0, β ∈ {0, 1} and xα y β ∈ L(n∞ D∞ ),
α
β y 1 |2α + β ≤ nQ , α ≥ 0, β ∈ {0, 1} and Q ∈ J ∩ Hr (Fq ), x−xQ x−xQ (gmQ )α (gβQ )|mα + β ≤ nQ , α ≥ 0, 0 ≤ β < m and Q ∈ J ∩ Hs (Fq ) y−hi (x) (x−xP )i
∈ F with deg hi (x) ≤ i − 1, vP (y − hi (x)) ≥ i, vP (y − ⎧ ⎨ −(2g + 1 − 2i), if d ≡ 1(mod2) hi (x)) = 0, vP (giP ) = −i and v∞ (giP ) ≥ ⎩ −(g + 1 − i), if d ≡ 0(mod2) (where v∞ (h) = Q∈supp (D∞ ) vQ (h)). α β So, by Lemma 2.3, it is possible that λ(x y ) = λ(x)α λ(y)β , for to prove
α
β
α
β λ(y) y 1 1 xα y β ∈ L(G). We also have λ = x−xQ x−xQ λ(x)−xQ λ(x)−xQ where giP :=
for 2α + β ≤ nQ , α ≥ 0, β ∈
α {0, 1} and Q ∈ J ∩ Hr (Fq ). Moreover λ(y)−pβ (λ(x)) m (λ(x)) λ((gmQ )α (gβQ )) = λ(y)−p for mα + β ≤ nQ , α ≥ 0, (λ(y)−xQ )i (λ(y)−xQ )i 0 ≤ β < m and Q ∈ J ∩ Hs (Fq ). Since every element h(x, y) ∈ L(G) can be written as a linear combination of the above elements, we proved that λ(h(x, y)) = h(λ(x), λ(y)). We will prove now that λ(x) and λ(y) satisfy the relation H(λ(x), λ(y)) = λ(y)2 − f (λ(x)) = d+1 0. We start noticing that x and x[ 2 ] belong to L(G) and so, by Lemma 2.3,
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t we have deg (λ(x))∞ ≤ g+1 ≤ 2td that is deg ((λ(x)d )∞ ) ≤ 2t < nN from which follows, since deg f (x) = d, that deg(f (λ(x)))∞ ≤ 2t. Moreover, since λ(y) ∈ L(G) we have deg (λ(y)2 )∞ = 2deg (λ(y))∞ ≤ 2t. So we have that λ(y)2 , f (λ(x)) ∈ L(2G). We also remark that λ(y)2 (P, φ) = f (λ(x))(P, φ) for each (P, φ) ∈ supp Φ, in fact: λ(y)2 (Pπ(i) , φπ(i) ) = φπ(i) (λ(y)2 + Pπ(i) ) = φπ(i) ((λ(y)+Pπ(i) )2 ) = (φπ(i) (λ(y)+Pπ(i) ))2 = (λ(y)(Pπ(i) , φπ(i) ))2 = (y(Pi , φi ))2 = y 2 (Pi , φi ) = f (x)(Pi , φi ) = φi (f (x) + Pi ) = φi (f (x + Pi )) = f (φi (x + Pi )) = f (x(Pi , φi )) = f (λ(x)(Pπ(i) , φπ(i) )) = f (φπ(i) (λ(x) + Pπ(i) )) = φπ(i) (f (λ(x) + Pπ(i) )) = φπ(i) (f (λ(x)) + Pπ(i) ) = f (λ(x))(Pπ(i) , φπ(i) ). Then, since deg 2G < nN , by Lemma 2.2, we have λ(y)2 = f (λ(x)). It is easy to show that the : Fq (x, y) → Fq (x, y) such that λ(x) = λ(x) and λ(y) = λ(y) is a map λ to L(G) Fq −endomorphism. It is obvious moreover that the restriction of λ is coincides with λ and so, since x and y are in the image of λ (x, y ∈ L(G)), λ ∈ Aut(F |Fq )Φ,G . But, an automorphism of F |Fq . We have only to prove that λ since G > 0 and λ(L(G)) = L(G), it follows that (see [11]) λ(G) = G and so it is possible to apply Lemma 2.4 from which follows that λ(Pi φi ) = (Pπ(i) , φπ(i) ) that is λ(Φ) = Φ. 2
The next theorem shows that similar result can be also obtained in the rational case. Theorem 2.6 Let Fq (x)|Fq be a rational function field. Let Φ = N i=1 (Pi , φi ) be a φ−divisorwith Pi pairwise distinct places all of the same degree n. Let G = rP∞ + m i=1 ri Pαi with r > 0, ri ≥ 0 and Pαi rational places. Let C(Φ; G; n) be the GAG−code associated with Φ and G. If deg G < nN and 2 nN ≥ 3 then H(Φ; G; n) ∼ Aut(F (x)|F ) . = q q Φ,G Now we give some constructions of rational GAG−codes and of their n−automorphism groups. Let Fq (x)|Fq be a rational function field. Let P = Pp(x) be a place of degree n 1 (where p(x) is a monic, irreducible polynomial of degree n). Let α ∈ Fq be a fixed root of p(x) (Fq denotes the algebraic closure of the finite field Fq ). 2 n−1 It is well-known that α, αq , αq , . . . , αq are exactly the n roots of p(x). So, one of its roots identifies clearly the polynomial p(x) and so, without creating ambiguity, it is possible to indicate Pp(x) with P[α] . With abuse of notation we will indicate also P∞ with P[∞] . Let FP = FPp(x) be the residue class field of P . There are exactly n Fq −isomorphisms between this field and the field Fqn . In fact, the map for all z + P = u(x) + P ∈ FP φα : FP → Fqn such that φα (z + P ) := u(α) v(α) v(x) is, clearly, a field Fq −isomorphism and the n Fq −isomorphisms between the
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(i)
fields FP and Fqn will be φα := ρi φα for i = 0, 1, . . . , n − 1, where ρ is the Frobenius’s automorphism of Fqn |Fq . (i)
qi
(i)
(i)
) , we have φα = φαqi and then (P[α] , φα ) = (P[αqi ] , φαqi ). Since φα (z+P ) = u(α v(αqi ) This means that, if we choose properly the root, we can always suppose a φ−place to be of the type (P[α] , φα ). It is well-known how the projective group P GL(2, q) acts on the projective line P G(1, q) = Fq ∪{∞}. Whereas its action on Fq (x)|Fq is defined as follows.
−1 f (σ[A] (x)) (x) (x) := g(σ−1 Let σ[A] ∈ P GL(2, q) and fg(x) ∈ Fq (x) then σ[A] fg(x) where (x)) [A] ⎛ ⎞ ab −1 dx−b ⎠ ∈ GL(2, q). σ[A] (x) = −cx+a if A = ⎝ cd It is well-known also that every projectivity acts like an Fq −automorphism of Fq (x) and that Aut(Fq (x)|Fq ) ∼ = P GL(2, q) (see for instance [6]). When σ[A] will denote an automorphism of Fq (x)|Fq it will be denoted with σA . Note that, since Fq ⊆ Fqn , we have that P GL(2, q) is, up to isomorphism, GL(2, q n ) ∼ a subgroup of P GL(2, q n ) (and so Aut(Fq (x)|Fq ) ∼ = P GL(2, q)⊆P = Aut(Fqn (x)|Fqn )). Thus P GL(2, q) acts on Fqn too. In fact, if σ[A] ∈ P GL(2, q) and α ∈ Fqn , we have that σ[A] (α) := aα+b if α ∈ / Fq , while σ[A] (α) is defined cα+d like usual if α ∈ Fq . Let’s remark that if α ∈ / Fq (and so n > 1) then cα+d = 0 and so σ[A] (α) is well defined. Finally, we can describe the action of Aut(Fq (x)|Fq ) on the set of φ−places of Fq (x)|Fq .
Proposition 2.7 If (P[α] , φα ) is a φ−place of Fq (x)|Fq and σA ∈ Aut(Fq (x)|Fq ) then σA (P[α] , φα ) = (P[σ[A] (α)] , φσ[A] (α) ). We will describe now the stabilizer Aut(Fq (x)|Fq )Φ,kP∞ and, with this, we can give a different version of the Theorem 2.6. Proposition 2.8 Let Φ = N i=1 (P[αi ] , φαi ) be a φ−divisor with P[αi ] distinct places all of degree n > 1 of the function field Fq (x)|Fq . Let G = rP∞ . Then the automorphisms σA ∈ Aut(Fq (x)|Fq ) which fix Φ and G are the affinities of the affine line which permute the roots αi . Corollary 2.9 Let Fq (x)|Fq be a rational function field and let C(Φ; G; n) be nN the GAG−code associated with Φ = N i=1 (P[αi ] , φαi ) and G = rP∞ . If r < 2 then H(n; Φ; G) is the group of the affinities of the affine line which permute the roots αi .
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