Weight enumerators of Reed-Muller codes from cubic curves and their duals

Let $\mathbb{F}_q$ be a finite field of characteristic not equal to $2$ or $3$. We compute the weight enumerators of some projective and affine Reed-Muller codes of order $3$ over $\mathbb{F}_q$. These weight enumerators answer enumerative questions about plane cubic curves. We apply the MacWilliams theorem to give formulas for coefficients of the weight enumerator of the duals of these codes. We see how traces of Hecke operators acting on spaces of cusp forms for $\operatorname{SL}_2(\mathbb{Z})$ play a role in these formulas.Comment: 19 pages. To appear in "Arithmetic, Geometry, Cryptography, and Coding Theory" (Y. Aubry, E. W. Howe, C. Ritzenthaler, eds.), Contemp. Math., 201

p-Adic estimates of Hamming weights in Abelian codes over Galois rings

A generalization of McEliece's theorem on the p-adic valuation of Hamming weights of words in cyclic codes is proved in this paper by means of counting polynomial techniques introduced by Wilson along with a technique known as trace-averaging introduced here. The original theorem of McEliece concerned cyclic codes over prime fields. Delsarte and McEliece later extended this to Abelian codes over finite fields. Calderbank, Li, and Poonen extended McEliece's original theorem to cover cyclic codes over the rings /spl Zopf//sub 2//sup d/, Wilson strengthened their results and extended them to cyclic codes over /spl Zopf//sub p//sup d/, and Katz strengthened Wilson's results and extended them to Abelian codes over /spl Zopf//sub p//sup d/. It is natural to ask whether there is a single analogue of McEliece's theorem which correctly captures the behavior of codes over all finite fields and all rings of integers modulo prime powers. In this paper, this question is answered affirmatively: a single theorem for Abelian codes over Galois rings is presented. This theorem contains all previously mentioned results and more

Traces of hecke operators and refined weight enumerators of reed-solomon codes

We study the quadratic residue weight enumerators of the dual projective Reed-Solomon codes of dimensions 5 and q − 4 over the finite field Fq. Our main results are formulas for the coefficients of the the quadratic residue weight enumerators for such codes. If q = p v and we fix v and vary p then our formulas for the coefficients of the dimension q − 4 code involve only polynomials in p and the trace of the q th and (q/p2 ) th Hecke operators acting on spaces of cusp forms for the congruence groups SL2(Z), Γ0(2), and Γ0(4). The main tool we use is the Eichler-Selberg trace formula, which gives along the way a variation of a theorem of Birch on the distribution of rational point counts for elliptic curves with prescribed 2-torsion over a fixed finite field

Weights of irreducible cyclic codes

With any fixed prime number p and positive integer N, not divisible by p, there is associated an infinite sequence of cyclic codes. In a previous article it was shown that a theorem of Davenport-Hasse reduces the calculation of the weight distributions for this whole sequence of codes to a single calculation (essentially that of calculating the weight distribution for the simplest code of the sequence). The primary object of this paper is the development of machinery which simplifies this remaining calculation. Detailed examples are given. In addition, tables are presented which essentially solve the weight distribution problem for all such binary codes with N < 100 and, when the block length is less than one million, give the complete weight enumerator