C. Fernandez-Cordoba | Searcharxiv

arXiv2009

Self-Dual Codes over Z_2xZ_4

Self-dual codes over $\Z_2\times\Z_4$ are subgroups of $\Z_2^α\times\Z_4^β$ that are equal to their orthogonal under an inner-product that relates to the binary Hamming scheme. Three types of self-dual codes are defined. For each type, the possible values $α,β$ such that there exist a code $\C\subseteq \Z_2^α\times\Z_4^β$ are established. Moreover, the construction of a $\add$-linear code for each type and possible pair $(α,β)$ is given. Finally, the standard techniques of invariant theory are applied to describe the weight enumerators for each type.

cs.IT↗

arXiv2009

Propelinear structure of Z_{2k}-linear codes

Let C be an additive subgroup of $\Z_{2k}^n$ for any $k\geq 1$. We define a Gray map $Φ:\Z_{2k}^n \longrightarrow \Z_2^{kn}$ such that $Φ(\codi)$ is a binary propelinear code and, hence, a Hamming-compatible group code. Moreover, $Φ$ is the unique Gray map such that $Φ(C)$ is Hamming-compatible group code. Using this Gray map we discuss about the nonexistence of 1-perfect binary mixed group code.

cs.IT↗