Abstract :
In Langevin and Zanotti (1995), we introduced a new class of codes called balanced weight distribution (BWD)-codes, with the remarkable property that their weight distribution is balanced, i.e., there are the same number of codewords for each non-zero weight. The aim of this paper is to study the weights of such codes in the irreducible cyclic case. First we recall the fundamental property of BWD-codes, and we start this study from a deep link between weights and Gauss sums. We see that this particular situation is, roughly speaking, in opposition to those studied by McEliece (1974). We give the weights for the two-weight case, and we show that the weights ofanyN-weight BWD-code defined over qis completely determined by theNweights of a BWD-code of dimensionNdefined over the prime field p. The main result is on the asymptotic behavior of Gauss sums over p, by means of a nice technique introduced by Rodier (1993).
