ON THE INTEGRAL RING SPANNED BY GENUS TWO WEIGHT ENUMERATORS. Manabu Oura

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1 ON THE INTEGRAL RING SPANNED BY GENUS TWO WEIGHT ENUMERATORS Manabu Oura Abstract. It is known that the weight enumerator of a self-ual oublyeven coe in genus g = 1 can be uniquely written as an isobaric polynomial in certain homogeneous polynomials with integral coefficients. We settle the case where g = 2 an prove the non-existence of such polynomials uner some conitions. 1. Introuction. In this paper we eal with binary self-ual oublyeven coes only. We refer to [3] for the general facts on coing theory. We shall first recall our problem in the case where g = 1, which explains what this paper concerns about. It is known that the weight enumerator of any self-ual oubly-even coe can be uniquely written as an isobaric polynomial in ϕ 8 = x 8 +14x 4 y 4 +y 8 an ϕ = x 4 y 4 (x 4 y 4 ) 4 with integral coefficients ([5]). We note that ϕ itself is not the weight enumerator of a coe but a linear combination of the weight enumerators with rational coefficients. We shall a a few wors on this basis. We consier the elements in Z[x, y] for simplicity. The choice of ϕ 8 is unique (up to ±1) since there exists a unique self-ual oubly-even coe e 8 of length 8. Next we assume that another homogeneous polynomial ξ of egree has the property in question, i.e., the weight enumerator of any self-ual oubly-even coe can be written as an isobaric polynomial in ϕ 8 an ξ with integral coefficients. We put ξ = ax + bx 20 y 4 +, a, b Z, in which the unwritten part consists of terms of egree less than 20 in x. There are 85 classes selfual oubly-even coes of length ([1], [2]) an the weight enumerator of these classes shoul be written as mϕ nϕ 8 ξ, in which m, n are integers. Examining these conitions for all classes, we know that 42a + b must be a ivisor of 1. We have that ξ = aϕ 8 ± ϕ an conversely, such ξ has the sai property. In the rest of this paper we restrict ourselves to the case where g = 2 when consiering the weight enumerators. Let C be a binary self-ual oublyeven coe an W C = W C (x, y, z, w) the weight enumerator of C in genus 2 (cf. [6], [4], [7]). We remark that W C is symmetric in x, y, z, w. We shall enote by W the grae ring over the fiel C of complex numbers generate by W C of all self-ual oubly-even coes. The egree -part W of W is a finite imensional vector space over C. The four elements 1

2 W e8, W +, W g, W + are algebraically inepenent over C an the grae ring W is a free C[W e8, W +, W g, W + ]-moule with a basis 1, W +, where g is the extene Golay coe of length. The imension formula of this ring is (im W ) t 1 + t = (1 t 8 )(1 t ) 2 (1 t ) 0 = 1 + t 8 + t t + 4t + 5t + 8t t We always keep this formula in min through the next section. 2. Result. For the proof of our Theorem, we shall construct homogeneous polynomials X 8, X, Y, X, X of egrees 8,,,,, respectively. This is one by analyzing the vector spaces W, = 8,,,. (egree 8) The extene Hamming coe e 8 of length 8 is a unique self-ual oubly-even coe of this length. We put X 8 = W e8. X 8 is also characterize by x 8 +. (egree ) Two polynomials X, Y are characterize by 0x + x 20 (y 4 + ) + 0x 18 (y 2 z 2 w 2 ) +, 0x + 0x 20 (y 4 + ) + x 18 (y 2 z 2 w 2 ) +. As we remarke, the weight enumerator in this paper is symmetric an x 20 (y 4 + ) stans for x 20 (y 4 + z 4 + w 4 ). We note that 0 as a coefficient of x 18 (y 2 z 2 w 2 ) in the first formula is not much of importance. The general form of the elements in W is a 0 x + a 1 x 20 (y 4 + ) + a 2 x 18 (y 2 z 2 w 2 ) + an is uniquely written as a 0 X8 3 + ( 42a 0 + a 1 )X + ( 504a 0 + a 2 )Y. (egree ) The polynomial X is characterize by 0x + 0x 28 (y 4 + ) + 0x 26 y 2 z 2 w 2 + x (y 4 z 4 + ) +. We remark that 0x + 0x 28 (y 4 + ) + implies that the coefficient of x (y 8 + ) is 0. The similar remark also hols in the following (egree ). The general form of the elements in W is a 0 x +a 1 x 28 (y 4 + )+a 2 x 26 (y 2 z 2 w 2 )+x ( a 3 (y 8 + ) + a 4 (y 4 z 4 + ) ) + 2

3 an is uniquely written as a 0 X8 4 +( 56a 0 +a 1 )X 8 X +( 672a 0 +a 2 )X 8 Y +(784a 0 33a 1 2a 2 +a 4 )X, where a 3 = 620a a 1. (egree ) The polynomial X is characterize by 0x +0x 36 (y 4 + )+0x 34 (y 2 z 2 w 2 )+0x (y 4 z 4 + )+x 28 (y 4 z 4 w 4 )+. The general form of the elements in W is a 0 x + a 1 x 36 (y 4 + ) + a 2 x 34 (y 2 z 2 w 2 ) + x (a 3 (y 8 + ) + a 4 (y 4 z 4 + )) +a 5 x 30 (y 6 z 2 w 2 + ) + x 28 (a 6 (y 12 + ) + a 7 (y 8 z 4 + ) + a 8 (y 4 z 4 w 4 )) + an is uniquely written as a 0 X ( 70a 0 + a 1 )X 2 8 X + ( 8a 0 + a 2 )X 2 8 Y + (1960a 0 61a 1 2a 2 + a 4 )X 8 X +(196560a a 1 880a a 4 + a 8 )X, where we have the relations a 3 = 285a 0 +a 1, a 5 = 84a 1 8a 2 +12a 4, a 6 = 21280a a 1, a 7 = 225a 1 + a a 4. The homogeneous polynomials we have obtaine can be written as X 8 = W e8, X = We W W g, Y = We W W g, X = We W e8 W W e8 W g W +, X = W 5 e W 2 e 8 W We 2 8 W g W e8 W W +. We note that X 8, X, Y, X, X are in Z[x, y, z, w] an that they generate the ring W. These being prepare, we prove Theorem. There exist no five homogeneous polynomials of egrees 8,,,, in W Z[x, y, z, w] such that the weight enumerator of any self-ual oubly-even coe can be written as an isobaric polynomial in these five elements with integral coefficients. Proof. Suppose that there exist such homogeneous polynomials of egrees 8,,,, satisfying the property in Theorem. As we iscusse in this 3

4 section, any element in W Z[x, y, z, w] of egree at most equal to can be uniquely written as an isobaric polynomial in X 8, X, Y, X, X with integral coefficients an the five assume polynomials are hence integral polynomials in X 8,..., X. Therefore X 8,..., X also enjoy the property in Theorem, i.e., the weight enumerator of any self-ual oubly-even coe can be written as i,j,k,l,m Z 0 a ijklm X i 8X j Y k X l X m, in which all a ijklm are integers. The weight enumerator of the coe + 56 is, however, written as X X 4 8 X X 4 8 Y X 3 8 X X 2 8 X X 8 X X 8 X Y X 8 Y X X Y X. This expression is unique an we get a contraiction. This completes the proof of Theorem. If we take a self-ual oubly-even coe C of length 48, an write W C as an isobaric polynomial in X 8, X, Y, X, X, then we can show that the coefficients in this expression are in Z[ 1 3 ]. It was, therefore, expecte to fin a counter example to our assumption in the proof of Theorem at this length, but it i not work out that way. Acknowlegement. The author woul like to thank Professor Nebe for her comments on this manuscript. This research was partially supporte by the Ministry of Eucation, Science, Sports an Culture, Grant-in-Ai for Young Scientists (B). References [1] Conway, J. H., Pless, V., On the enumeration of self-ual coes, J. Combin. Theory Ser. A 28 (1980), no. 1, [2] Conway, J. H., Pless, V., Sloane, N. J. A., The binary self-ual coes of length up to : a revise enumeration, J. Combin. Theory Ser. A 60 (1992), no. 2,

5 [3] Conway, J. H., Sloane, N. J. A., Sphere packings, lattices an groups, Thir eition, Grunlehren er Mathematischen Wissenschaften 290, Springer-Verlag, New York, [4] Duke, W., On coes an Siegel moular forms, Internat. Math. Res. Notices 1993, no. 5, [5] Gleason, A. M., Weight polynomials of self-ual coes an the MacWilliams ientities, Actes u Congrès International es Math maticiens (Nice, 1970), Tome 3, pp Gauthier-Villars, Paris, [6] Huffman, W. C., The biweight enumerator of self-orthogonal binary coes, Discrete Math. 26 (1979), no. 2, [7] Nebe, G., Rains, E.M., Sloane, N.J.A., Self-ual coes an invariant theory, Algorithms an Computation in Mathematics 17, Springer- Verlag, Berlin,