On Some New $\mathbb{Z}_{m}$ Linear Codes Based on Elementary Symmetric Functions

Abstract

Let $\mathbb{Z}_{m}\overset{\text{def}}{=}\{0,1, \ldots, (m-1)\}$ be the m-ary alphabet, $m\in \mathbb{N}$ . This paper gives some new theory and efficient designs of $\mathbb{Z}_{m}$ linear error control codes based on the elementary symmetric functions of m-ary words. Here, a $\mathbb{Z}_{m}$ linear code is a sub-module of the module $(\mathbb{Z}_{m}^{n}, +\text{mod} m, \mathbb{Z}_{m}, \cdot \text{mod} m), n\in \mathbb{N}$ , and the errors are measured in the $L_{1}$ or Lee metric. In particular, given a field, K, of characteristic $p$ = char(K) = $2, 3,5, \ldots$ prime, and given $d, m=vp^{l}, v, l, n\in \mathbb{N}$ with $d\leq m/v=p^{l}$ and $n\leq\vert K\vert -1$ , we introduce a new class of (d-1) asymmetric error correcting $\mathbb{Z}_{m}$ linear codes, $\mathcal{C}_{d}$ , of length $n$ whose redundancy is only $\rho(\mathcal{C}_{d})=n-\log_{m}\vert\mathcal{C}_{d}\vert \leq(d-1)\log_{m}\vert K\vert$ . For these codes we give very efficient field based algebraic decoding algorithms to control $d - 1$ errors actually in the Lee distance. Also for the extended codes, we give new efficient field based decoding algorithms.