Duality Preserving Gray Maps for Codes over Rings

Given a finite ring $A$ which is a free left module over a subring $R$ of $A$, two types of $R$-bases, pseudo-self-dual bases (similar to trace orthogonal bases) and symmetric bases, are defined which in turn are used to define duality preserving maps from codes over $A$ to codes over $R$. Both types of bases are generalizations of similar concepts for fields. Many illustrative examples are given to shed light on the advantages to such mappings as well as their abundance

New extremal binary self-dual codes of length 68 via short kharaghani array over f_2 + uf_2

In this work, new construction methods for self-dual codes are given. The methods use the short Kharaghani array and a variation of it. These are applicable to any commutative Frobenius ring. We apply the constructions over the ring F_2 + uF_2 and self-dual Type I [64, 32, 12]_2-codes with various weight enumerators obtained as Gray images. By the use of an extension theorem for self-dual codes we were able to construct 27 new extremal binary self-dual codes of length 68. The existence of the extremal binary self-dual codes with these weight enumerators was previously unknown.Comment: 10 pages, 5 table

Codes and the Steenrod algebra

We study codes over the finite sub Hopf algebras of the Steenrod algebra. We define three dualities for codes over these rings, namely the Eulidean duality, the Hermitian duality and a duality based on the underlying additive group structure. We study self-dual codes, namely codes equal to their orthogonal, with respect to all three dualities