Definition et Caracterisation d'une Dimension Minimale pour les Codes Principaux Nilpotents d'une Algebre Modulaire de p-Groupe Abelien Elementaire

L'ensemble des idéaux de L'algèbre est partitionné en sous-ensembles déterminés a partir de la situation d'un idéal dans la suite décroissante d'idéaux que forment les codes de Reed et Muller Généralisés (GRM-codes). Dans chaque sous-ensemble, la dimension des idéaux est bornée inférieurement. Nous caractérisons les idéaux de dimension minimale; nous en déduisons une nouvelle représentation des éléments du GRM-code d'ordre 1.The set of the ideals belonging to the algebra is divided into subsets; they are determined by the place of an ideal in the decreasing series of ideals composed by the Generalized Reed and Muller codes (GRM-codes). A lower bound is obtained for the dimension of the ideals in each subset. We characterize the minimal dimension ideals and such investigation permits us to represent elements of the first order GRM-code

Self-dual codes which are principal ideals of the group algebra "F2[{F2m,+}]"

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Crooked functions

International audienceCrooked permutations were introduced twenty years ago since they allow to construct interesting objects in graph theory. The field of applications was extended later. Crooked functions, bijective or not, correspond to APN functions and to some optimal codes. We adopt an unified presentation of crooked functions, explaining the connection with partially-bent functions. We then complete some known results and derive new properties. For instance, we observe that crooked functions allow to construct sets of bent functions and define some permutations

Crooked functions

Crooked permutations were defined twenty years ago. It was firstly shown that they can be used to construct interesting objects in graph theory. The field of applications was extended later, since crooked functions, bijective or not, correspond to APN functions and to some optimal codes. We adopt an unified presentation, of crooked functions, explaining the connexion with partially-bent functions. We then complete some known results and propose new properties. For instance, crooked functions allow to construct sets of bent functions, or simply define some permutations

1-translated codes of a generalized Reed and Muller code

A coset of a generalized Reed and Muller code (GRM-code) which is between two successive GRM-codes is called 1-translated. The GRM-codes are here characterized as the powers of an A-modular-algebra radical. The steadiness of the 1-translated codes over some A algebra automorphisms points their algebraic properties .Un translaté d'un code de Reed et Muller généralisé (GRM-code) qui est compris entre deux GRM-codes successifs est dit 1-translaté. Les GRM-codes sont ici considérés comme les puissances du radical d'une algèbre modulaire A. La stabilité des codes 1-translatés sous certains automorphismes de l'algèbre A spécifie les propriétés algébriques de ces codes

Tools for Coset Weight Enumerators of some Codes

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Weight distributions of cosets of 2-error-correcting binary BCH codes, extended or not

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The crooked property

International audienceCrooked permutations were introduced twenty years ago to cons- truct interesting objects in graph theory. These functions, over F2n with odd $n$, are such that their derivatives have as image set a com- plement of a hyperplane. The field of applications was extended later, in particular to cryptography. However binary crooked functions are rare. It is still unknown if non quadratic crooked functions do ex- ist. We extend the concept and propose to study the crooked property for any characteristic. A function $F$, from Fpn to itself, satisfies this property if all its derivatives have as image set an a ne subspace. We show that the partially-bent vectorial functions and the functions satisfying the crooked property are strongly related. We later focus on the components of these functions, establishing that the existence of linear structures is here decisive. We then propose a symbolic ap- proach to identify the linear structures. We claim that this problem consists in solving a system of linear equations, and can often be seen as a combinatorial problem

Cyclic codes with few weights and Niho exponents

AbstractThis paper studies the values of the sums Sk(a)=∑x∈F2m(-1)Tr(xk+ax),a∈F2m,where Tr is the trace function on F2m, m=2t and gcd(2m-1,k)=1. We mainly prove that when k≡2j(mod2t-1), for some j, then Sk(a) takes at least four values when a runs through F2m. This result, and other derived properties, can be viewed in the study of weights of some cyclic codes and of crosscorrelation function of m-sequences