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On The Weights of Binary Irreducible Cyclic Yves Aubry and Philippe Langevin
 

Summary: On The Weights of Binary Irreducible Cyclic
Codes
Yves Aubry and Philippe Langevin
Universit´e du Sud Toulon-Var, Laboratoire GRIM F-83270 La Garde, France,
{langevin,yaubry}@univ-tln.fr,
WWW home page: http://{langevin,yaubry}.univ-tln.fr
Abstract. This 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 mul-
tiplicity of the weights. Secondly, in connection with a Schmidt and
White's conjecture, we focus on binary irreducible cyclic codes of in-
dex 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 imag-
inary 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

  

Source: Aubry, Yves - Institut de Mathématiques de Toulon, Université du Sud Toulon -Var

 

Collections: Mathematics