On Some Zm Linear Goppa/BCH like Error Control Codes and Elementary Symmetric Functions*

Let ${\mathbb{Z}_m}\mathop = \limits^{{\text{ def }}}$ $\left\{ {0,1, \ldots ,\left( {m - 1} \right)} \right\}$ be the m-ary alphabet, $m \in \mathbb{N}$. This paper gives some new theory and 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 submodule of the module $\left( {\mathbb{Z}_m^n, + {\text{mod}}m,{\mathbb{Z}_m}, \cdot {\text{mod}}m} \right),n \in \mathbb{N}$, and the errors are measured in the ${L_1}$ or Lee metric. Potentially, the alphabet size, m, can be any natural, however, the described code designs and decoding methods are solely based on fields and field operations. In particular, starting from a very general class of Goppa-like ${\mathbb{Z}_m}$ linear codes, given a field, $K$, of characteristic $p = {\text{char}}\left( K \right) \in \mathbb{N}$, we consider a generalization of the ${\mathbf{BCH}}$ codes to the m-ary alphabet for $m = {p^l},l \in \mathbb{N}$. For these BCHlike codes we are able to prove a BCH-like bound with respect to both the ${L_1}$ and Lee distances. This enabled us to design a wide family of remarkable efficient codes. For example, an efficient design is given for ${\mathbb{Z}_m}$ linear codes with $m = {2^l},l \in \mathbb{N}$, length $n = m$, minimum Lee distance ${d_{Lee}} = m = n$ and the number of information m-ary digits $k = m/2 = n/2$.