Jacobi polynomials and design theory II

In this paper, we introduce some new polynomials associated to linear codes over $\mathbb{F}_{q}$. In particular, we introduce the notion of split complete Jacobi polynomials attached to multiple sets of coordinate places of a linear code over $\mathbb{F}_{q}$, and give the MacWilliams type identity for it. We also give the notion of generalized $q$-colored $t$-designs. As an application of the generalized $q$-colored $t$-designs, we derive a formula that obtains the split complete Jacobi polynomials of a linear code over $\mathbb{F}_{q}$.Moreover, we define the concept of colored packing (resp. covering) designs. Finally, we give some coding theoretical applications of the colored designs for Type~III and Type~IV codes.Comment: 28 page

Fonctions harmoniques, codes et designs

Cette thèse se compose de trois parties, chacune formée d'un chapitre de rappels et d'un chapitre contenant des "nouveautés". Dans la première partie, après des rappels sur la représentation du groupe symétrique, nous étudions certaines fonctions harmoniques associées à une représentation classique de Sn. Dans la deuxième partie nous donnons une caractérisation de certains designs généralisés dans le cadre des schémas d'association. Les fonctions harmoniques de la première partie nous permettent de déduire un algorithme pour tester si un ensemble donné est un design. Enfin dans la dernière partie nous définissons pour les codes binaires des énumérateurs de poids harmoniques ; dans le cas des codes autoduaux nous établissons des résultats d'invariance, notamment une identité de type MacWilliams.This thesis consists of three parts. In the first one, after some classical results about the representation of the symmetric group we study some harmonic functions attached to a representation of Sn. In the second part we characterize some generalized designs in the setting of association schemes. The harmonic functions computed in the first part allow us to derive an algorithm to test if a set is a design. In the last part we define some multiple harmonic enumerators of codes, associated to a binary code and to an harmonic function ; in the case of self-dual codes we prove for them a MacWilliams type equality

Tricolore 3-designs in Type III codes

A split complete weight enumerator in six variables is used to study the 3-colored designs held by codewords of fixed composition in Type III codes containing the allone codeword. In particular the ternary Golay code contains 3-colored 3-designs. We conjecture that every weight class in a Type III code with the all-one codeword holds 3-colored 3-designs. 1 Introduction Colored designs were introduced in [6] to study the Harada designs held by the lifted Golay over Z 4 : They produce ordinary designs by bleaching and are easily understood in terms of split (complete) weight enumerators. Colored t\Gammadesigns are known to exist when the permutation part of the automorphism group is t\Gammatransitive [6]. This is the case, for instance, of the doubled Golay code in [7], which is invariant under the action of M 24 : In the present work, we investigate the tricolore 3\Gammadesigns held by codewords of extremal Type III codes. It is well-known that the ternary Golay is invariant under the..

Tricolore 3-designs in Type III codes

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