On The Weights of Irreducible Cyclic Codes

The paper is devoted to the study of the weight distribution of irreducible cyclic codes. We start from the interpretation due to McEliece of these weights by means of Gauss sums. Firstly, a p-adic analysis using Stickelberger congruences and Gross-Koblitz formula enable us to improve the divisibility theorem of McEliece by giving results on the multiplicity of the weights. Secondly, in connection with a conjecture of Schmidt and White, we focus on index 2 binary irreducible cyclic codes. We show, assuming the generalized Riemann hypothesis, that there are infinitely many such codes. Furthermore, we consider a subclass of this family of codes, called Quadratic Residue codes. The parameters of these codes are related to class numbers of some imaginary quadratic number fields. We give an elementary proof of the non-existence of two-weight binary irreducible cyclic codes of index 2.