2,290 research outputs found
p-Adic valuation of weights in Abelian codes over /spl Zopf/(p/sup d/)
Counting polynomial techniques introduced by Wilson are used to provide analogs of a theorem of McEliece. McEliece's original theorem relates the greatest power of p dividing the Hamming weights of words in cyclic codes over GF (p) to the length of the smallest unity-product sequence of nonzeroes of the code. Calderbank, Li, and Poonen presented analogs for cyclic codes over /spl Zopf/(2/sup d/) using various weight functions (Hamming, Lee, and Euclidean weight as well as count of occurrences of a particular symbol). Some of these results were strengthened by Wilson, who also considered the alphabet /spl Zopf/(p/sup d/) for p an arbitrary prime. These previous results, new strengthened versions, and generalizations are proved here in a unified and comprehensive fashion for the larger class of Abelian codes over /spl Zopf/(p/sup d/) with p any prime. For Abelian codes over /spl Zopf//sub 4/, combinatorial methods for use with counting polynomials are developed. These show that the analogs of McEliece's theorem obtained by Wilson (for Hamming weight, Lee weight, and symbol counts) and the analog obtained here for Euclidean weight are sharp in the sense that they give the maximum power of 2 that divides the weights of all the codewords whose Fourier transforms have a specified support
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
Divisibility of Weil Sums of Binomials
Consider the Weil sum , where is
a finite field of characteristic , is the canonical additive
character of , is coprime to , and . We say that
is three-valued when it assumes precisely three distinct values as
runs through : this is the minimum number of distinct values in the
nondegenerate case, and three-valued are rare and desirable. When
is three-valued, we give a lower bound on the -adic valuation of
the values. This enables us to prove the characteristic case of a 1976
conjecture of Helleseth: when and is a power of ,
we show that cannot be three-valued.Comment: 11 page
Proof of a Conjectured Three-Valued Family of Weil Sums of Binomials
We consider Weil sums of binomials of the form , where is a finite field, is
the canonical additive character, , and .
If we fix and and examine the values of as runs
through , we always obtain at least three distinct values unless
is degenerate (a power of the characteristic of modulo ).
Choices of and for which we obtain only three values are quite rare and
desirable in a wide variety of applications. We show that if is a field of
order with odd, and with , then
assumes only the three values and . This
proves the 2001 conjecture of Dobbertin, Helleseth, Kumar, and Martinsen. The
proof employs diverse methods involving trilinear forms, counting points on
curves via multiplicative character sums, divisibility properties of Gauss
sums, and graph theory.Comment: 19 page
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