The Binary GOLAY Code Obtained from an Extended Cyclic Code over F8

The binary image of an extended cyclic code over F8 relatively to a self-dual basis of F8 is the GOLAY code (24, 12, S)

Asymptotically Good Additive Cyclic Codes Exist

Long quasi-cyclic codes of any fixed index $>1$ have been shown to be asymptotically good, depending on Artin primitive root conjecture in (A. Alahmadi, C. G\"uneri, H. Shoaib, P. Sol\'e, 2017). We use this recent result to construct good long additive cyclic codes on any extension of fixed degree of the base field. Similarly self-dual double circulant codes, and self-dual four circulant codes, have been shown to be good, also depending on Artin primitive root conjecture in (A. Alahmadi, F. \"Ozdemir, P. Sol\'e, 2017) and ( M. Shi, H. Zhu, P. Sol\'e, 2017) respectively. Building on these recent results, we can show that long cyclic codes are good over \F_q, for many classes of $q$'s. This is a partial solution to a fifty year old open problem

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

Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page