Survey on Z₂Z₄-additive codes

Abstract

A code C is Z₂Z₄-additive if the set of coordinates can be partitioned into two subsets X and Y such that the punctured code of C by deleting the coordinates outside X (respectively, Y ) is a binary linear code (respectively, a quaternary linear code). The corresponding binary codes of Z₂Z₄-additive codes under an extended Gray map are called Z₂Z₄-linear codes, which seem to be a very distinguished class of binary group codes. As for binary and quaternary linear codes, for these codes the fundamental parameters are shown and standard forms for generator and parity-check matrices are given, defining the appropriate concept of duality. The main results on Z₂Z₄-additive self-dual and Z₂Z₄-additive formally self-dual codes are also presented, as well as, the results on the invariants rank and dimension of the kernel for these codes are given. Several families of important binary codes fall in the class of Z₂Z₄-linear codes. In this survey, we review characterizations, properties and constructions of perfect and extended perfect Z₂Z₄-linear codes, Hadamard Z₂Z₄-linear codes, Reed-Muller Z₂Z₄-linear codes, maximum distance separable Z₂Z₄-linear codes, and Preparata-like and Kerdock-like Z₂Z₄-linear codes. Finally, applications of Z₂Z₄-additive codes to steganography are also presented.