This paper presents a study of perturbations of symmetric Boolean functions. In particular, it establishes a connection between exponential sums of these perturbations and Diophantine equations of the form Σl=0n(nl)xl= 0, where xjbelongs to some fixed bounded subset Γ of Z. The concepts of trivially balanced symmetric Boolean function and sporadic balanced Boolean function are extended to this type of perturbations. An observation made by Canteaut and Videau for symmetric Boolean functions of fixed degree is extended. To be specific, it is proved that, excluding the trivial cases, balanced perturbations of fixed degree do not exist when the number of variables grows. Some sporadic balanced perturbations are presented. Finally, a beautiful but unexpected identity between exponential sums for perturbations of two different symmetric Boolean functions is also included in this work.

In this paper we present an algorithmic approach to the problem of the divisibility of the number of solutions to a system of polynomial equations. Using this method we prove that all binary cyclic codes with two zeros over F2fand minimum distance 5 are quasi-perfect for f les 10. We also present elementary proofs of divisibility results that, in some cases, improve previous results

In this correspondence, we correct an error in the correspondence cited in the title.

This paper describes an application of our generalization of McEliece's theorem in order to improve Serre's bound. Serre obtained sharp estimates for the number of solutions of an algebraic curve over a finite field. This improvement depends of the divisibility properties of the curve. Curves with many rational points are important in algebraic-geometry codes and low-discrepancy sequences

We are presenting a new method to obtain the covering radius of codes and in particular to prove quasi-perfection in codes. Our techniques combine divisibility results of Ax-Katz and Moreno-Moreno as well as coding theoretic methods. We answer a problem posed by Cohen-Honkala-Litsyn-Lobstein in the book covering radius for Bose-Chaudhuri-Hocquenghem (BCH) codes. We also obtain the covering radius for many new classes of codes.