Different aspects of q-ary images of codes

This paper is a survey on results concerning q-ary images of codes over Fq m mainly about cyclic codes, doubly even binary self dual codes and applications to decoding.Cet article fait le point sur des résultats obtenus concernant la démultiplication des codes sur F qm (image q-aire) principalement à propos des codes cycliques, de codes autoduaux binaires à poids multiples de quatre et d'applications au décodage

On the weights of binary irreducible cyclic codes

International audienceThis paper is devoted to the study of the weights of binary irreducible cyclic codes. We start from McEliece's interpretation of these weights by means of Gauss sums. Firstly, a dyadic analysis, using the Stickelberger congruences and the Gross-Koblitz formula, enables us to improve McEliece's divisibility theorem by giving results on the multiplicity of the weights. Secondly, in connection with a Schmidt and White's conjecture, we focus on binary irreducible cyclic codes of index two. We show, assuming the generalized Riemann hypothesis, that there are an infinite of such codes. Furthermore, we consider a subclass of this family of codes satisfying the quadratic residue conditions. The parameters of these codes are related to the class number of some imaginary quadratic number fields. We prove the non existence of such codes which provide us a very elementary proof, without assuming G.R.H, that any two-weight binary irreducible cyclic code c(m,v) of index two with v prime greater that three is semiprimitive