OBSERVATION ON THE WEIGHT ENUMERATORS FROM CLASSICAL INVARIANT THEORY. By Manabu Oura 1

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1 OBSERVATION ON THE WEIGHT ENUMERATORS FROM CLASSICAL INVARIANT THEORY By Manabu Oura 1 The purpose of this paper is to present some relationships among invariant rings. This is done by combining two maps the Broué Enguehard map and Igusa s ρ homomorphism. For the completeness of the story some formulae and statements are given which are not necessarily needed in the present manuscript. Sections 1 and 2 have some expository nature and contain no new result. 1. Classical Invariant Theory. In this section we recall classical invariant theory. For the detail we refer to [1]. We consider a homogeneous polynomial n ( n u i x n i y i i i=0 of degree n in 2 variables x y. The group SL(2 C operates on the variable space and if we require that the above form is invariant the same group operates on the coefficient space. In this way we get an irreducible representation of SL(2 C of degree n + 1. We consider the graded ring of polynomials in the u 0 u 1... u n with coefficients in C and operate SL(2 C on this graded ring using its action on its homogeneous part of degree one defined by the above representation. Then the invariant subring S(2 n is a graded integrally closed integral domain over C. In the present paper we deal with S(2 4 and S(2 6. The structure theorems of those rings are established in the 19th century and we shall describe them. The invariant ring S(2 4 is generated by P Q which are algebraically independent. The explicit forms are and the dimension formula of S(2 4 is P = u 0 u 4 4u 1 u 3 + 3u 2 2 u 0 u 1 u 2 Q = det u 1 u 2 u 3 u 2 u 3 u 4 1 (1 t 2 (1 t 3 = 1 + t2 + t 3 + t 4 + t 5 + 2t 6 + t 7 + 2t + 2t 9 + 2t t t 12 + in which the coefficient of t k denotes the dimension of degree k-part of S(2 4. The invariant ring S(2 6 is generated by J 2 J 4 J 6 J 10 J 15. We give the definitions of J 2... J 15 in the appendix taken from [1]. Among them J 2 J 4 J 6 J 10 are algebraically independent. The ring C[J 2 J 4 J 6 J 10 ] contains J15 2 but not J 15. We also give the explicit formula for J15 2 in the appendix. The dimension formula of S(2 6 is given by 1 + t 15 (1 t 2 (1 t 4 (1 t 6 (1 t 10 = 1 + t2 + 2 t 4 + 3t 6 + 4t + 6t 10 + t t 14 + t t 16 + t t 1 + 2t t t 21 + t t t + 6t t 26 + t t t t This work is supported in part by KAKENHI(No

2 2. Weight Enumerators and Siegel Modular Forms. In this section we recall coding theory and Siegel modular forms. For the details we refer to [1] [15] for coding theory and to [3] for Siegel modular forms. Let F 2 be the field of two elements. A linear code (a code for short of length n is a subspace of F n 2. The vector space Fn 2 equips with the inner product x y = x i y i. We define the weight wt(v of a vector v F n 2 by the number of the non-zero coordinates of v. We shall define special classes of codes. If a code C coincides with its dual code C = {x F n 2 (x y = 0 y C} it is called self-dual. We observe that the dimension of the self-dual code is a half of its length. If the weight of any element in C is divisible by 4 it is called doubly-even. In this manuscript we will focus on the self-dual doubly-even codes. We shall next define a homogeneous polynomial of the code which is on the title of this paper. The weight enumerator W (g C of the code C in genus g is defined by W (g C = W (g C (x a : a F g 2 = v 1...v g a F g 2 x na(v1...vg a where n a (v 1... v g denotes the number of i such that a = (v 1i... v gi. If we need the ordering of the elements of F g 2 we fix Fg 2 = { }. We sometimes use the symbols x y z... instead of the x a s for simplicity. The weight enumerator in genus 1 is interpreted as W (1 C = v C x n wt(v y wt(v where n denotes the length of the code C. In this case the weight enumerator of a self-dual doubly-even code is a symmetric polynomial in the variables x y. The examples are W (1 e = x + 14x 4 y 4 + y W (1 = x + 759x 16 y x 12 y x y 16 + y where e denotes the extended Hamming code and the extended Golay code. We omit the definitions of e as well as d + n appearing below and refer to the references cited above. In the case when g = 2 the weight enumerator of a self-dual doubly-even code is also symmetric in the variables x y z w. We have W (2 e = ( + 14( ( W (2 = ( + 759( ( ( ( ( ( ( where ( = x + y + z + w ( = x 12 y 4 z 4 w 4 + x 4 y 12 z 4 w 4 + x 4 y 4 z 12 w 4 + x 4 y 4 z 4 w 12 etc. For an arbitrary positive integer n n 0 (mod we have = ( 1 α β x α+γ x α W (2 d + n βγ F 2 2 α F 2 2 We note that for g 3 the weight enumerator of a self-dual doubly-even code is not symmetric in general. We shall next view the weight enumerator from invariant theory of some finite group. Let H g (g 1 be a finite subgroup of GL(2 g C generated by ( g 1 + i ( ( ( 1 2 a b diag i S[a] ; a F g 2 S = t S Mat g g (Z ab F g 2 n/2. 2

3 where A[B] = t BAB for matrices A B of suitable sizes. H g has a normal subgroup N g = Z/4Z 2 1+2g + such that H g /N g = Sp(g F2 where denotes the central product and 2 1+2g + the extra special 2-group of order 1+2g of + type. The finite group H g has an order 2 g2 +2g+2 (4 g 1(4 g 1 1 (4 1. We define another finite group G g which is generated by H g and the primitive eighth root of unity. The group G g contains H g as a subgroup of index 2. We have defined two finite groups so far. The group which directly concerns the weight enumerators is G g. Indeed the weight enumerator of any self-dual doubly-even code is invariant under the action of G g. Moreover the invariant ring of G g can be generated by the weight enumerators of the self-dual doubly-even codes for any g (cf. [4] [6] [2] [5] [17] [11] [21]. Therefore we may regard the invariant ring C[x a ; a F g 2 ]Gg as the ring of weight enumerators of the self-dual doubly-even codes in genus g. Igusa s homomorphism is under some condition one from the ring A(Γ g of Siegel modular forms to the ring S(2 2g + 2 of projective invariants of a binary form of degree 2g + 2 (see [7]. We recall that that the ring A(Γ g is the graded ring generated by holomorphic functions ψ on the Siegel upper-half space S g in genus g satisfying the functional equation ψ(m τ = det(cτ + d k ψ(τ for every M in Γ g = Sp(g Z (plus a condition at infinity for g = 1. In order to construct Siegel modular forms we introduce the theta-constants. Theta-constants θ m m (τ are defined by θ m m (τ = ( ] 1 exp 2π 1 [p 2 τ + m + (p t + m m p Z g in which m and m are the column vectors in R g. If we put f m (τ = θ m 0(2τ the Broué- Enguehard map T h is defined by x a f a. A modular form is called a cusp form if it is in the kernel of Siegel s Φ-operator which maps a modular form of genus g to a modular form of genus g 1. The structures of the invariant rings and of Siegel modular forms in small genera are known. We shall describe the cases g = 1 2. First suppose that g = 1. The groups H 1 G 1 are finite unitary reflection groups of order respectively (No. No.9 in the list of [19]. We have where C[x y] H1 = C[W (1 e h (1 12 ] C[x y]g1 = C[W (1 e W (1 ] h (1 12 = x12 33x y 4 33x 4 y + y 12. The dimension formulae of these invariant rings are given by 1 (1 t (1 t 12 1 (1 t (1 t. The map T h induces the isomorphisms 2 C[x y] H1 = A(Γ 1 and C[x y] G1 = A(Γ 1 (4. Here we remark that A(Γ 1 = A(Γ 1 (2. The invariant ring C[x y] G1 is a subring of C[x y] H1 and we observe that W (1 = (W (1 e (h ( If S is a graded ring composed of homogeneous parts S k with k running over non-negative integers and if d is a positive integer then S (d = k 0 S dk. 3

4 The isomorphisms above are given by T h(w e (1 = φ 4 (ω T h(h (1 12 = φ 6(ω T h(w (1 = (φ 4 (ω (φ 6 (ω 2 where φ k (ω denotes the normalized Eisenstein series of weight k: φ k (ω = 1 +. We shall next discuss the case when g = 2. The group H 2 is a finite unitary reflection group of order 4600 No.31 in the list of [19]. The invariant ring of H 2 is generated by the four elements W e (2 W g (2 h (2 12 = (12 33( ( ( F 20 = (20 19( ( ( ( ( ( ( ( The dimension formula of this ring is 1 (1 t (1 t 12 (1 t 20 (1 t = 1+t +t 12 +t 16 +2t 20 +3t +2t 2 +4t 32 +4t 36 +5t The group G 2 contains H 2 by index 2 and is not a finite unitary reflection group. The invariant ring is generated by W e (2 W g (2 W (2 W (2 W (2 W (2 d + d + 32 and W (2. The four elements W W (2 40 are algebraically independent and the square of W (2 is written by the polynomial in W W g (2 32 W (2 W (2 as follows: d + d + 40 (W (2 2 = (W (W e (2 5 W g ( (W e (2 5 W (2 d (W e (2 4 W (2 d (W e (2 3 W (2 d (W (2 e 2 (W ( (W e (2 2 W g (2 W (2 d (W e (2 2 (W (2 2 d W (2 e W (2 W (2 d W ( W (2 W (2 d W (2 d + W (2 d e W (2 W (2 d + d + 32 d + d + 40 This was given in [21]. The dimension formula of this invariant ring C[x y z w] G2 is 1 + t 32 (1 t (1 t 2 (1 t 40 = 1 + t + t t + 4t t 40 + t t t

5 The elements W (2 W (2 W (2 can be written by the generators of C[x y z w] H2 d + d + 32 d + 40 as follows: W (2 = (W (h ( W g (2 W (2 = (W W e (2 (h ( W e (2 W g ( h (2 12 F 20 W (2 = (W (W e (2 2 (h ( (W e (2 2 W g ( W e (2 h (2 12 F F20 2. We give a comment on the paper [9]. In that paper Maschke determined the invariant ring of some finite group G. G is a subgroup of SL(4 C and has an order 4600 which is the same as our H 2. G is a subgroup of our G 2 which is of an order = H 2 is generated by three elements ( i ( ( 1 2 a b ab F 2 2 and G 2 by H 2 and 1+i 2 while G is generated by diag ( diag ( ( i ( ( 1 a b 2 ab F i 2 diag ( i 2 diag ( The dimension formula of C[x y z w] G is given by 1 + t 32 + t 60 + t 92 (1 t (1 t 2 (1 t 40. From the dimension formulae for example we can read off the differences among the invariant rings of the said groups. We continue our discussion on our case. We shall recall that A(Γ 2 is generated over C by five elements and they are ψ 4 = (θ m 2 2 ψ 6 = ± (θ m1 θ m2 θ m3 4 syzygous 2 14 χ 10 = (θ m χ 12 = (θ m1 θ m2 θ m ( ( 1 χ 35 = θm azygous ± (θ m1 θ m2 θ m3 20 In the second symmetrization the monomial (θ m1 θ m2 θ m3 4 with t m 1 = ( t m 2 = ( t m 3 = ( has +1 as its coefficient. In the definition of χ 12 the summation is extended over fifteen complements of syzygous quadruples. In the definition of χ 35 the symmetrization of ± (θ m1 θ m2 θ m3 20 is taken by the stabilizer of θ m in Sp(2 Z modulo. 3 χ 35 is not used in Section 3. 5

6 the stabilizer of (θ m1 θ m2 θ m3 20 with t m 1 = ( t m 2 = ( t m 3 = ( The Broué-Enguehard map gives rise the following: T h(w (2 e = ψ 4 T h(h (2 12 = ψ 6 T h(f 20 = ψ 4 ψ χ 10 T h(w (2 = ψ ψ χ 12. These can be obtained by comparing the Fourier coefficients (cf. [16] [14] [13]. There have been extensive studies on Fourier coefficients of Siegel modular forms however in our case we do not need a deep theory of Fourier coefficients. Since there is a misprint in the definition of F 4 in [] (corrected in [10] we reproduce the formulae which are useful for our computations of Fourier coefficients. In the case when g = 1 we shall use ω instead of τ. If we put F 0 (r = r p2 F 1 (r = r (p 1/22 p=1 in which r = exp π 1ω then we have θ 00 (ω = 1 + 2F 0 (r θ 01 (ω = 1 + 2F 0 ( r θ 10 (ω = 2F 1 (r. In the case when g = 2 if we put F 0 (r 1 r 2 = F 0 (r 1 + F 0 (r 2 + F 1 (r 1 r 2 = F 1 (r 2 + p 1p 2=1 p=1 p 1p 2=1 A p1p 2 r p2 1 1 rp2 2 2 B p1p 2 r p2 1 1 r(p2 1/22 2 F 2 (r 1 r 2 = F 1 (r 2 r 1 F 3 (r 1 r 2 = C p1p 2 r (p1 1/22 1 r (p2 1/22 2 F 4 (r 1 r 2 = p 1p 2=1 p 1p 2=1 D p1p 2 r (p1 1/22 1 r (p2 1/22 2 in which r 1 = exp π 1τ 1 r 2 = exp π 1τ 2 q 12 = exp 2π 1τ 12 and A p1p 2 = q p1p q p1p2 12 B p1p 2 = q p1(p2 1/ q p1(p2 1/2 12 C p1p 2 = q (p1 1/2(p2 1/ q (p1 1/2(p2 1/2 12 D p1p 2 = ( 1 p1+p2 1 q (p1 1/2(p2 1/ ( 1 p1 p2 q (p1 1/2(p2 1/2 12 6

7 then we will have θ 0000 (τ = 1 + 2F 0 (r 1 r 2 θ 0001 (τ = 1 + 2F 0 (r 1 r 2 θ 0010 (τ = 1 + 2F 0 ( r 1 r 2 θ 0011 (τ = 1 + 2F 0 ( r 1 r 2 θ 0100 (τ = 2F 1 (r 1 r 2 θ 0110 (τ = 2F 1 ( r 1 r 2 θ 1000 (τ = 2F 2 (r 1 r 2 θ 1001 (τ = 2F 2 (r 1 r 2 θ 1100 (τ = 2F 3 (r 1 r 2 θ 1111 (τ = 2F 4 (r 1 r 2. Cusp forms can be written by Eisenstein series as follows: χ 10 = (ψ 4 ψ 6 ψ 10 χ 12 = ( ψ ψ ψ 12. We note that there is a misprint in the formula of χ 10 at p.102 in [16]. 3. The Broué-Enguehard map and Igusa s homomorphism. Before proceeding to Igusa s homomorphism studied in [7] we go back to the invariant rings S(2 4 S(2 6. addition to the generators of them given in Section 1 we give the different generators in irrational forms. If we decompose a binary form into linear factors as n i=0 ( n u i x n i y i = u 0 i n i=1 (x ξ i y In then we have ( n 1 u1 u 0 = n ξ i i=1 ( n u2 = ( n ξ i ξ j ( 1 n un = 2 u 0 n u i<j 0 n ξ i. i=1 Preparing this we consider each case separately 4. We put P 2g+2 (x = u 0 (x ξ 1 (x ξ 2 (x ξ 2g+2. Suppose that g = 1. In [7] Igusa takes I 2 (P 4 (x I 3 (P 4 (x as the generators of S(2 4 given as follows: I 2 (P 4 (x = u 2 0 (12 2 (34 2 three = u 2 { 0 (12 2 ( (13 2 ( 2 + (14 2 (23 2} I 3 (P 4 (x = u 3 0 (12 2 (34 2 (13( six = u 3 { 0 (12 2 (34 2 {(13( + (14(23} + (13 2 ( 2 {(12(34 (14(23} + (14 2 (23 2 { (12(34 (13(} }. Here (ij is an abridged notation for ξ i ξ j. We already gave the generators of S(2 4 in Section 1 and these two sets of the generators are related each other by I 2 (P 4 (x = 2 3 3P I 3 (P 4 (x = Q. 4 We note that the case when g = 1 is roughly sketched in [12]. 7

8 Igusa s homomorphism ρ gives the isomorphism ρ : A(Γ 1 = S(2 4 where ρ(ψ 4 (τ = 2 1 I 2 (P 4 (x ρ(ψ 6 (τ = 2 1 I 3 (P 4 (x. Combining two maps T h and ρ to denote ρ we have the isomorphism C[x y] H1 by = S(2 4 given If we use P Q defined in Section 1 then ρ(w e (1 = 2 1 I 2 (P 4 (x ρ(h (1 12 = 2 1 I 3 (P 4 (x. ρ(w (1 e = 2 2 3P ρ(h (1 12 = Q. We also get C[x y] G1 = S(2 4 (2 where ρ(w (1 = I 2 (P 4 (x I 3 (P 4 (x 2 = ( 11P Q 2. We shall next consider the case when g = 2. In [7] the following elements 5 are used as the generators of S(2 6. A(P 6 (x = u 2 0 (12 2 (34 2 (56 2 fifteen B(P 6 (x = u 4 0 (12 2 (23 2 (31 2 (45 2 (56 2 (64 2 ten C(P 6 (x = u 6 0 (12 2 (23 2 (31 2 (45 2 (56 2 (64 2 (14 2 (25 2 (36 2 sixty D(P 6 (x = u 10 0 (122 (13 2 (56 2 }{{} E(P 6 (x = u 15 0 fifteen There hold the following relations. A(P 6 (x = J 2 B(P 6 (x = ( J J 4 fifteen 1 ξ 1 + ξ 2 ξ 1 ξ 2 det 1 ξ 3 + ξ 4 ξ 3 ξ 4. 1 ξ 5 + ξ 6 ξ 5 ξ 6 C(P 6 (x = ( J J 2 J J 6 D(P 6 (x = ( J J 3 2 J J 2 2 J J 2 J J 4 J J 10 E(P 6 (x = J We note that we do not need E in this paper.

9 We do not need the formula of E 2 in this paper however since it is not contained in [7] we give the explicit formula of E 2. This is derived from the formula of J15 2 in the appendix or directly. E 2 = (1/ ( A 7 B A 6 B 3 C A 6 B 2 D A 5 B A 5 B 2 C A 5 BCD A 5 D A 4 B 4 C A 4 B 3 D A 4 BC A 4 C 2 D 3 53A 3 B A 3 B 3 C A 3 B 2 CD A 3 BD A 3 C A 2 B 5 C A 2 B 4 D A 2 B 2 C A 2 BC 2 D A 2 CD AB AB 4 C AB 3 CD AB 2 D ABC AC 3 D 2 7 3B 6 C B 5 D B 3 C B 2 C 2 D BCD C D 3. Igusa used this to get the expression for χ 2 35 in [7]. Igusa s ρ-homomorphism is given by ρ(ψ 4 = 2 2 B ρ(ψ 6 = 2 3 (AB 3C = ( J J 2 J J 6 ρ(χ 10 = 2 14 D ρ(χ 12 = AD = ( J J 4 2 J J 3 2 J J 2 2 J J 2 J 4 J J 2 J 10 ρ(χ 35 = D 2 E. This homomorphism is injective (Theorem 5 in [7]. If we shall denote by ρ the composition of the Broué-Enguehard map and Igusa s ρ-homomorphism we will have ρ(w (2 e = ρ(ψ 4 = 2 2 B ρ(h (2 12 = ρ(ψ 6 = 2 3 (AB 3C ρ(f 20 = ρ(ψ 4 ψ χ 10 = 2 5 ( AB 2 3BC D = 3 7 ( J J 3 2 J J 2 2 J J 2 J J 4 J J 10 ρ(w (2 = ρ( ψ ψ χ 12 = ( 7A 2 B ABC AD + 11B C 2 = ( J J 4 2 J J 3 2 J J 2 2 J J 2 J 4 J J 2 J J J

10 Using these formulae we know the ρ image of the generators (except W (2 e W (2 of C[x y z w] G2 as follows: ρ(w (2 = ρ( (W (h ( W g (2 = ( 2A 2 B ABC AD + 11B C 2 = 3 4 5( J J 4 2 J J 3 2 J J 2 2 J J 2 J 4 J J 2 J J J 2 6 ρ(w (2 = ρ( (W W e (2 (h ( W e (2 W g ( h (2 12 F 20 = ( 2 4 A 2 B AB 2 C ABD + 43B BC CD = 3 5 5( J J 6 2J J 5 2 J J 4 2 J J 3 2 J 4J J 3 2 J J 2 2 J J 2 2 J J 2 J 2 4 J J 2 J 4 J J J 4 J J 6 J 10 ρ(w (2 = ρ( (W (W e (2 2 (h ( (W e (2 2 W g ( W e (2 h (2 12 F F20 2 = ( 2 5A 2 B AB 3 C AB 2 D + 19B B 2 C BCD D 2 = 3 7 5( J J 2 J J 7 2 J J 6 2 J J 5 2J 4 J J 5 2 J J 4 2 J J 4 2 J J 3 2 J 2 4 J J 3 2J 4 J J 2 2J J 2 2J 4 J J 2 2 J 6J J 2 J 3 4 J J 2 J 2 4 J J J 2 4 J J 4 J 6 J J We observe that the ρ images of A(Γ 2 (2 A(Γ 2 (4 are strictly smaller than S(2 6 (2 S(2 6 (4 respectively. On the other hand it is known that the Broué-Enguehard map induces the isomorphisms C[x y z w] H2 = A(Γ 2 (2 and C[x y z w] G2 = A(Γ 2 (4. Therefore the ρ images of the rings C[x y z w] H2 C[x y z w] G2 Summing up are strictly smaller than S(2 6 (2 S(2 6 (4 respectively. Theorem. Let ρ be the composition of the Broué-Enguehard map and Igusa s ρ-homomorphism. (1 In the case when g = 1 ρ gives rise to the isomorphisms from C[x y] H1 onto S(2 4 and from C[x y] G1 onto S(2 4 (2. (2 In the case when g = 2 ρ transforms injectively C[x y z w] H2 and C[x y z w] G2 into S(2 6. The ρ images of these invariant rings are strictly smaller than S(2 6 (2 S(2 6 (4 respectively. The explicit ρ-images of the generators of each invariant ring are given above in two ways each. 10

11 We give remarks. (1 Since Igusa s ρ homomorphism increases the weight or the degree by a 1 2g ratio our ρ increases the degree by a 1 4g ratio. This remark holds in the arbitrary genus g. Here we note that Siegel modular forms we are considering are always of even weights. (2 The weight enumerator of a code has non-negative integers as its coefficients(in the arbitrary genus. We shall consider the case when g = 1. The ρ image of the weight enumerator has negative coefficients as the polynomials in C[u 0 u 1 u 2 u 3 u 4 ] in general. For example we have ρ(w (1 e = 2 2 3(u 0 u 4 4u 1 u 3 + 3u 2 2 ρ(w (1 = ( 11u 3 0u u 2 0u 1 u 3 u u 2 0u 2 2u u 2 0u 2 u 2 3u u 2 0u u 0 u 2 1u 2 u u 0 u 2 1u 2 3u 4 36u 0 u 1 u 2 2u 3 u 4 756u 0 u 1 u 2 u 3 3 1u 0 u 4 2 u u 0 u 3 2 u u4 1 u u3 1 u 2u 3 u 4 704u 3 1 u u 2 1u 3 2u u 2 1u 2 2u u 1 u 4 2u u 6 2. It would be interesting if we interpret this from coding theoretical or combinatorial point of view. (3 We mention the paper [20] in which Shioda discussed the close relationship of the ring S(3 4 of projective invariants to the invariant theory for the Weyl groups W (E 7 and W (E 6. We omit the details. Acknowledgement. The author would like to thank Prof.Tsuyumine for helpful discussion. Sapporo Medical University REFERENCES. [1] Conway J. H. Sloane N. J. A. Sphere packings lattices and groups 3rd edition Grundlehren der Mathematischen Wissenschaften 290 Springer-Verlag New York(1999. [2] Duke W. On codes and Siegel modular forms Int. Math. Res. Notices No. 5 ( [3] Freitag E. Siegelsche Modulfunktionen Grundlehren der Math. Wiss. vol 254 Berlin Heidelberg New York: Springer (193. [4] Gleason A. M. Weight polynomials of self-dual codes and the MacWilliams identities Actes Congrès Intern. Math. 3 ( [5] Herrmann N. Höhere Gewichtszäher von Codes und deren Beziehung zur Theorie der Siegelschen Modulformen Diplomarbeit Universität Bonn [6] Huffman W. C. The biweight enumerator of self-orthogonal binary codes Discrete Math. 26(1979 no [7] Igusa J. Modular forms and projective invariants Amer. J. Math. 9 (

12 [] Igusa J. On the ring of modular forms of degree two over Z Amer. J. Math. 101 (1979 no [9] Maschke H. Ueber die quaternäre endliche lineare Substitutionsgruppe der Borchardt schen Moduln Math. Ann. 30( [10] Nagaoka S. On the ring of Hilbert modular forms over Z J. Math. Soc. Japan Vol. 35 (193 No [11] Nebe G. Rains E. M. Sloane N. J. A. The invariants of the Clifford groups Des. Codes Cryptogr. ( [12] Oura M. Note on the weight enumerators J. of Liberal Arts and Sciences Sapporo Medical University School of Medicine. [13] Ozeki M. On basis problem for Siegel modular forms of degree 2 Acta Arith. 31 (1976 no [14] Ozeki M. Washio T. An extended table of the Fourier coefficients of Eisenstein series of degree two Bull. Fac. Liberal Arts Nagasaki Univ. 23 (192/3 no [15] Rains E. M. Sloane N. J. A. Self-dual codes in Handbook of coding theory ed. by Pless V. S. et al Amsterdam Elsevier (199. [16] Resnikoff H. L. Saldaña R. L. Some properties of Fourier coefficients of Eisenstein series of degree two J. Reine Angew. Math. 265 ( [17] Runge B. Codes and Siegel modular forms Discrete Math. 14 ( [1] Schur I. Vorlesungen über Invariantentheorie Die Grundlehren der mathematischen Wissenschaften 143 Springer-Verlag Berlin Heidelberg New York 196. [19] Shephard T. C. Todd J. A. Finite unitary reflection groups Can. J. Math (1954. [20] Shioda T. Some new observation on invariant theory of plane quartics Kodaira s issue Asian J. Math. 4 (2000 no [21] Vardi I. Coding theory (Multiple weight enumerators of codes preprint

13 Appendix. We give the generators of S(2 6 from [1]. We also give the expression for J J 2 = u 0 u 6 6u 1 u u 2 u 4 10u 2 3 u 0 u 1 u 2 u 3 J 4 = det u 1 u 2 u 3 u 4 u 2 u 3 u 4 u 5 u 3 u 4 u 5 u 6 b 0 b 1 b 2 J 6 = det b 1 b 2 b 3 b 2 b 3 b 4 J 10 = u 0 c 3 6u 1 bc 2 + 3u 2 (ac + 4b 2 c 4u 3 (3abc + 2b 3 + 3u 4 a(ac + 4b 2 6u 5 a 2 b + u 6 a 3 c 0 c 1 c 2 c 3 c 4 c 1 c 2 c 3 c 4 c 5 J 15 = det c 2 c 3 c 4 c 5 c 6 c 3 c 4 c 5 c 6 c 7 c 4 c 5 c 6 c 7 c where b 0 = 6(u 0 u 4 4u 1 u 3 + 3u 2 2 b 1 = 3(u 0 u 5 3u 1 u 4 + 2u 2 u 3 b 2 = u 0 u 6 9u 2 u 4 + u 2 3 b 3 = 3(u 1 u 6 3u 2 u 5 + 2u 3 u 4 b 4 = 6(u 2 u 6 4u 3 u 5 + 3u 2 4 a = 2(u 0 u 2 u 6 3u 0 u 3 u 5 + 2u 0 u 2 4 u 2 1u 6 + 3u 1 u 2 u 5 u 1 u 3 u 4 3u 2 2u 4 + 2u 2 u 2 3 b = u 0 u 3 u 6 u 0 u 4 u 5 u 1 u 2 u 6 u 1 u 3 u 5 + 9u 1 u u2 2 u 5 17u 2 u 3 u 4 + u 3 3 c = 2(u 0 u 4 u 6 u 0 u 2 5 3u 1 u 3 u 6 + 3u 1 u 4 u 5 + 2u 2 2u 6 u 2 u 3 u 5 3u 2 u u 2 3u 4 c 0 = (u 2 0 u 5 5u 0 u 1 u 4 + 2u 0 u 2 u 3 + u 2 1 u 3 6u 1 u 2 2 c 1 = u 2 0u 6 + 2u 0 u 1 u 5 19u 0 u 2 u 4 + u 0 u 2 3 6u 2 1u u 1 u 2 u 3 30u 3 2 c 2 = 2(u 0 u 1 u 6 2u 0 u 2 u 5 2u 0 u 3 u 4 3u 1 u 2 u u 1 u u2 2 u 3 c 3 = u 0 u 2 u 6 4u 0 u 3 u 5 2u 0 u u2 1 u 6 6u 1 u 2 u 5 + u 1 u 3 u 4 15u 2 2 u 4 c 4 = 4( u 0 u 4 u 5 + u 1 u 2 u 6 + 3u 1 u 2 4 3u 2 2u 5 c 5 = u 0 u 4 u 6 2u 0 u u 1u 3 u 6 + 6u 1 u 4 u 5 + 2u 2 2 u 6 u 2 u 3 u u 2 u 2 4 c 6 = 2( u 0 u 5 u 6 + 2u 1 u 4 u 6 + 2u 2 u 3 u 6 + 3u 2 u 4 u 5 16u 2 3u u 3 u 2 4 c 7 = u 0 u 2 6 2u 1u 5 u u 2 u 4 u 6 + 6u 2 u 2 5 u2 3 u 6 44u 3 u 4 u u 3 4 c = ( u 1 u u 2 u 5 u 6 2u 3 u 4 u 6 u 3 u u 2 4u 5. 13

14 J 2 15 = J J 13 2 J J 12 2 J J 11 2 J J 10 2 J 4 J J 10 2 J J 9 2 J J 9 2 J J 2 J 2 4 J J 2 J 4 J J 7 2 J J 7 2 J 4 J J 7 2 J 6J J 6 2 J 3 4 J J 6 2 J 2 4 J J 6 2 J J 5 2 J J 5 2 J 2 4 J J 5 2 J 4J 6 J J 5 2 J J 4 2 J 4 4 J J 4 2 J 3 4 J J 4 2 J 4J J 4 2 J 2 6 J J 3 2 J J 3 2 J 3 4 J J 3 2 J 2 4 J 6 J J 3 2 J 4J J 3 2 J J 2 2 J 5 4 J J 2 2 J 4 4 J J 2 2 J 2 4 J J 2 2 J 4 J 2 6 J J 2 2 J 6J J 2 J J 2 J 4 4 J J 2 J 3 4 J 6 J J 2 J 2 4 J J 2 J 4 J J 2 J 3 6 J J 6 4 J J 5 4 J J 3 4 J J 2 4 J 2 6 J J 4 J 6 J J J