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

Let Zm := {0,1,…,(m−1)} be the m-ary alphabet, m∈N. This paper gives some new theory and designs of Zm linear error control codes based on the elementary symmetric functions of m-ary words. Here, a Zm linear code is a submodule of the module (Znm,+modm,Zm,⋅modm),n∈N, and the errors are measured in the L1 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 Zm linear codes, given a field, K, of characteristic p=char(K)∈N, we consider a generalization of the BCH codes to the m-ary alphabet for m=pl,l∈N. For these BCHlike codes we are able to prove a BCH-like bound with respect to both the L1 and Lee distances. This enabled us to design a wide family of remarkable efficient codes. For example, an efficient design is given for Zm linear codes with m=2l,l∈N, length n=m, minimum Lee distance dLee=m=n and the number of information m-ary digits k=m/2=n/2.