Abstract: 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 crypto graphical 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

Keywords: Non-linearity, finite groups, characters, Fourier transforms, Reed-Muller codes, boolean functions, covering radius, cryptography.

Abstract: 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 been introduced in a previous paper and have the remarkable property that each non-zero weight appears with the same multiplicity. We show that an irreducible cyclic BWD-code with N non-zero weights defines an association scheme with N classes. We study such a code from two different point of vue: the code as a K-subspace of Kⁿ where K is a finite field, and the code through an extension [L : K] isomorphic to the code C. From the first one, the code is a subscheme of the Hamming scheme and from the second one, it is a cyclotomic scheme.

Keywords: Automorphism group, BWD-codes, cyclotomic schemes, Hamming schemes, irreducible cyclic codes.

Abstract: In a previous paper, we have introduced a new class of codes called BWD-codes, with a remarkable property, their weight distribution is balanced, i.e. there are the same number of codewords for each non-zero weight. The aim of this paper is to study the weights of such codes in the irreducible cyclic case. First we recall the fundamental property of BWD-codes and we start this study from a deep link between weights and Gauss sums. We see that this particular situation is, roughly speaking, at the opposite of those studied by McEliece. We give the weights for the two-weight case, and we show that the weights of any N-weight BWD-code defined over F_q is completely determined by the N-weights of a BWD-code of dimension N defined over the F_p. The main result concerns the asymptotic behavior of Gauss sums over the prime field F_p, by means of a nice technique introduced by Rodier.

Abstract: A In this paper, we study particular linear codes defined over F_q, with an astonishing property, their weight distribution is balanced, i.e. there is the same number of codewords for each nonzero weight of the code. We call these codes BWD-codes. We first study BWD-codes by means of the Pless identities and we completely characterize the two-weight projective case. We study the class of codes defined under subgroups of the multiplicative group of F_q^s using Gauss sums. Then, given a prime p and an integer N dividing p - 1, we construct all the N-weight BWD-codes of that class. We conclude this paper by some tables of BWD-codes and an open problem.

Keywords: BWD-codes, codes under groups, Gauss sums, irreducible cyclic codes, trace function, transposed codes.

Abstract:In this paper we give a new upper bound on the number of bent functions in 8 variables which is at most 2^129.2. First, and this is the main work in this paper, we find a partial classification of quartic forms in 8 variables under the action of GL(8,2), the general linear group over F₂. Once this partial classification is obtained, we are able to estimate the number of possible lower degree terms we can add to each representative by means of equations over their binary coefficients.

Keywords: Bent functions in 8 variables, Reed-Muller codes, cryptography, Schreier trees, orbits, general linear group, stabilizers, homogeneous polynomials, compound matrices.

Abstract: Following Patterson and Wiedemann, we find counterexamples for a conjecture of Mykkelveit related to the covering radius of the first order Reed-Muller code. One of them has a remarkable algebraic structure.

Keywords: Frobenius, Reed-Muller codes, covering radius. Walsh transform.

Résumé: L'objet de cet article est d'étudier les différentes techniques de codage qui entrent en jeu dans la réalisation du disque compact, codage à l'enregistrement et codage correcteur.

Mots Clefs: Disque compact, Code EFM, Code CIRC, codes de Reed-Solomon.

Abstract: Theoretical study of highly non-linear boolean functions, and balanced highly non-linear functions.

Abstract: Theoretical study of highly non-linear boolean functions, and balanced highly non-linear functions.

Résumé: Nous étudions principalement une nouvelle famille de codes cycliques irréductibles possédant une propriété remarquable, leur distribution de poids est équilibrée (codes BWD, Balanced Weight Distribution). Nous caractérisons ces codes et nous montrons que leurs poids ne dépendent que de sommes de Gauss évaluées en des caractères d'un groupe particulier. Nous étudions aussi leurs propriétés combinatoires remarquables, notamment nous montrons que ces codes définissent des schémas d'association cyclotomiques. Une étude spécifique des sommes de Gauss nous permet de calculer les poids sur toute extension de F_q à partir de la connaissance des poids du plus petit code sur le corps premier F_p. Nous explicitons aussi le comportement asymptotique des poids au regard de la borne de Weil-Serre dans ce sens où ces poids sont liés aux nombres de points rationnels de certaines courbes d'Artin-Schreier.

Mots Clefs: Formes coordonnées, codes C(f), codes cycliques irréductibles, distribution de poids, codes BWD, sommes de caractères, sommes de Gauss, transformée de Fourier, schémas d'association cyclotomiques, courbes d'Artin-Schreier, borne de Weil-Serre.

Abstract: We study a new class of irreducible cyclic codes with a remarkable property, their weight distribution is balanced (BWD-codes). We characterize these codes and we show that their weights depend only on certain Gauss-sums for characters defined over a particular group. We also study their remarkable combinatorial properties, especially, we show that these codes define cyclotomic association schemes. A specific study of related Gauss-sums allows us to compute the weights on any extension of F_q with the knowledge of the weights of the smallest code defined over the prime field F_p. We characterize the asymptotic behavior of weights compared to the Weil-Serre bound, in that sense the weights are related to the numbers of rational points of some Artin-Schreier curves.

Keywords: Coordinate forms,codes C( f ), irreducible cyclic codes, weight distribution, BWD-codes, character sums, Gauss sums, Fourier transform, cyclotomic association schemes, Artin-Schreier curves, Weil-Serre bound.