Partial Permutation Decoding for Several Families of Linear and <inline-formula> <tex-math notation="LaTeX">${\mathbb{Z}}_4$ </tex-math> </inline-formula>-Linear Codes

Abstract

A general criterion to obtain $s$ -PD-sets of minimum size $s+1$ for partial permutation decoding, which enable correction up to $s$ errors, for systematic codes over a finite field ${ {F}}_{q}$ and ${ {Z}}_{4}$ -linear codes is provided. We show how this technique can be easily applied to linear cyclic codes over ${ {F}}_{q}$ , ${ {Z}}_{4}$ -linear codes which are the Gray map image of a quaternary linear cyclic code, and some related codes such as quasi-cyclic codes. Furthermore, specific results for some linear and nonlinear binary codes, including simplex, Kerdock, Delsarte-Goethals, and extended dualized Kerdock codes are given. Finally, applying this technique, new $s$ -PD-sets of size $s+1$ for ${ {Z}}_{4}$ -linear Hadamard codes of type $2^\gamma 4^\delta $ , for all $\delta \geq 4$ and $1 ; and for ${ {Z}}_{4}$ -linear simplex codes of type $4^{m}$ , for all $m\geq 2$ and $1 , are also provided.