Daniel J. Katz

Weil sums of binomials, three-level cross-correlation, and a conjecture of Helleseth

Let q be a power of a prime p, let ?q:Fq?C be the canonical additive character ?q(x)=exp(2πiTrFq/Fp(x)/p), let d be an integer with gcd(d,q-1)=1, and consider Weil sums of the form Wq,d(a)=?x?Fq?q(xd+ax). We are interested in how many different values Wq,d(a) attains as a runs through Fq*. We show that if |{Wq,d(a):a?Fq*}|=3, then all the values in {Wq,d(a):a?Fq*} are rational integers and one of these values is 0. This translates into a result on the cross-correlation of a pair of p-ary maximum length linear recursive sequences of period q-1, where one sequence is the decimation of the other by d: if the cross-correlation is three-valued, then all the values are in Z and one of them is -1. We then use this to prove the binary case of a conjecture of Helleseth, which states that if q is of the form 22n, then the cross-correlation cannot be three-valued.

New open problems related to old conjectures by Helleseth

Recently, very interesting results have been obtained concerning the Fourier spectra of power permutations over a finite field. In this note we survey the recent ideas of Aubry, Feng, Katz, and Langevin, and we pose new open problems related to old conjectures proposed by Helleseth in the middle of the seventies.

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. McElieces 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 Zopf(2d) 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 Zopf(pd ) 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 Zopf(pd) with p any prime. For Abelian codes over Zopf 4, combinatorial methods for use with counting polynomials are developed. These show that the analogs of McElieces 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

Low Correlation Sequences From Linear Combinations of Characters

Pairs of binary sequences formed using linear combinations of multiplicative characters of finite fields are exhibited that, when compared with a random sequence pairs, simultaneously achieve significantly lower mean square autocorrelation values (for each sequence in the pair) and significantly lower mean square crosscorrelation values. If we define crosscorrelation merit factor analogously to the usual merit factor for autocorrelation, and if we define demerit factor as the reciprocal of merit factor, then randomly selected binary sequence pairs are known to have an average crosscorrelation demerit factor of 1. Our constructions provide sequence pairs with a crosscorrelation demerit factor significantly less than 1, and at the same time, the autocorrelation demerit factors of the individual sequences can also be made significantly less than 1 (which also indicates better than average performance). The sequence pairs studied here provide combinations of autocorrelation and crosscorrelation performance that are not achievable using sequences formed from single characters, such as maximal linear recursive sequences (m-sequences) and Legendre sequences. In this paper, exact asymptotic formulae are proved for the autocorrelation and crosscorrelation merit factors of sequence pairs formed using linear combinations of multiplicative characters. Data is presented that shows that the asymptotic behavior is closely approximated by sequences of modest length.

Aperiodic Crosscorrelation of Sequences Derived From Characters

It is shown that the pairs of maximal linear recursive sequences (m-sequences) typically have mean square aperiodic crosscorrelation on par with that of random sequences, but that if one takes a pair of m-sequences where one is the reverse of the other, and shifts them appropriately, one can get significantly lower mean square aperiodic crosscorrelation. Sequence pairs with even lower mean square aperiodic crosscorrelation are constructed by taking a Legendre sequence, cyclically shifting it, and then cutting it (approximately) in half and using the halves as the sequences of the pair. In some of these constructions, the mean square aperiodic crosscorrelation can be lowered further if one truncates or periodically extends (appends) the sequences. Exact asymptotic formulas for mean squared aperiodic crosscorrelation are proved for sequences derived from additive characters (including m-sequences and modified versions thereof) and multiplicative characters (including Legendre sequences and their relatives). Data are presented that show that the sequences of modest length have performance that closely approximates the asymptotic formulas.

p-Adic estimates of hamming weights in abelian codes over galois rings

A generalization of McElieces 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 McElieces original theorem to cover cyclic codes over the rings Zopf2 d, Wilson strengthened their results and extended them to cyclic codes over Zopf p d, and Katz strengthened Wilsons results and extended them to Abelian codes over Zopfp d. It is natural to ask whether there is a single analogue of McElieces 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

On theorems of Delsarte---McEliece and Chevalley---Warning---Ax---Katz

We present a theorem that generalizes the result of Delsarte and McEliece on the p-divisibilities of weights in abelian codes. Our result generalizes the Delsarte–McEliece theorem in the same sense that the theorem of N. M. Katz generalizes the theorem of Ax on the p-divisibilities of cardinalities of affine algebraic sets over finite fields. As the Delsarte–McEliece theorem implies the theorem of Ax, so our generalization implies that of N. M. Katz. The generalized theorem gives the p-divisibility of the t-wise Hamming weights of t-tuples of codewords (c(1), . . . ,c(t)) as these words range over a product of abelian codes, where the t-wise Hamming weight is defined as the number of positions i in which the codewords do not simultaneously vanish, i.e., for which

{\BBZ}/p^d{\BBZ}

{(c^{(1)}_i,\ldots,c^{(t)}_i)\not=(0,\ldots,0)}