Abnmct -In ISL J. Wolfmaun we6 a descrlplh ol projective blnary l l w v codes by means of a polynomW. 'Ibp purpose of our work Is fist to ieaerallze such a description to q w coder. Iben. the maln dlfllculty 1s to evaluate the mlghtr of these codes a d It In not possible to follow the same way Y In binary case.W e com$l&r Anla-sehreler curves and we five a bound on welghts.Some aormallzatloa are also Investlgated.Throughout the paper, k will designate a positive integer, q a power of a prime p , Fq-the finite field of order 4,, F; , = Fq \ (0) and s = 4r -1. We denote by RT[x] the set of reduced polynomials, i.e. R,[x] = Fq[x]/(xs -1). The trace function of FC over f, is denoted by tr.
One of the hardest problems in coding theory is to evaluate the covering radius of first order Reed-Muller codes RM(1, m), and more recently the balanced covering radius for cryptographical purposes. The aim of this paper is to present some new results on this subject. We mainly study boolean functions invariant under the action of some finite groups, following the idea of Patterson and Wiedemann [The covering radius of the (1, 15) Reed-Muller Code is atleast 16276, IEEE Trans Inform Theory, Vol. 29 (1983) 354.]. Our method is Fourier transforms and our results are both theoretical and numerical.
In this paper we characterize the automorphism group of a particular class of irreducible cyclic codes by means of an important theorem of Carlitz and McConnel. The BWD-codes (balanced weight distribution) have the remarkable property that each nonzero weight appears with the same multiplicity. We also show that an irreducible cyclic BWD-code with N nonzero weights is a subscheme with N classes of the Hamming scheme isomorphic to a cyclotomic scheme.1997 Academic Press
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