8 research outputs found
On a conjecture of Helleseth
We are concern about a conjecture proposed in the middle of the seventies by
Hellesseth in the framework of maximal sequences and theirs cross-correlations.
The conjecture claims the existence of a zero outphase Fourier coefficient. We
give some divisibility properties in this direction
On a Conjecture of Helleseth
Abstract. We are concerned about a conjecture proposed in the middle of the seventies by Hellesseth in the framework of maximal sequences and theirs cross-correlations. The conjecture claims the existence of a zero outphase Fourier coefficient. We give some divisibility properties in this direction
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
Cyclic codes with few weights and Niho exponents
AbstractThis paper studies the values of the sums Sk(a)=∑x∈F2m(-1)Tr(xk+ax),a∈F2m,where Tr is the trace function on F2m, m=2t and gcd(2m-1,k)=1. We mainly prove that when k≡2j(mod2t-1), for some j, then Sk(a) takes at least four values when a runs through F2m. This result, and other derived properties, can be viewed in the study of weights of some cyclic codes and of crosscorrelation function of m-sequences
On maximal period linear sequences and their crosscorrelation functions /
For an nth order linear recurring sequence over the finite field Fp. the largest possible period is pn --- 1. When such a sequence attains this upper bound as its period, it is called a maximal period linear sequence, or m-sequence in short. Interest in such sequences originated from applications. Indeed, there is an interaction between m-sequences, coding theory and cryptography via the relation with cyclic codes.Boolean functions, etc. One of the main goals is to construct a pair of binary m-sequences whose crosscorrelation takes few values, preferably with small magnitude. By a theorem of Helleseth. the crosscorrelation function takes at least three values.Hence, existence and construction of sequences with 3-valued crosscorrelation is of particular interest. This is also the main theme of our work. The aim of this thesis is to introduce foundational material on m-sequences, explain the relations with other topics mentioned above, and to present proofs of three conjectures on the existence/nonexistence of 3-valued crosscorrelation functions for binary m-sequences. These conjectures are due to Sarwate-Pursley, Helleseth and Welch and were proved by McGuire-Calderbank. Calderank-MeGnire-Poonen-Rubinstein and. Canteaut-Charpin-Dobbertin respectively