On Some New Zm Linear Codes Based on Elementary Symmetric Functions

Let Z(m) def= 0; 1, ... , (m - 1)g be the m-ary alphabet, mN. This paper gives some new theory and efficient designs of Z(m) linear error control codes based on the elementary symmetric functions of m-ary words. Here, a Z(m) linear code is a sub-module of the module (Z(m)(n);+ mod m; Zm; . mod m), n is an element of 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, ... prime, and given d,m = vp(l), v, l, nN with d <= m/v = p(l) and n <= vertical bar K vertical bar - 1, we introduce a new class of (d -1) asymmetric error correcting Z(m) linear codes, C-d, of length n whose redundancy is only p(C-d) = n - log(m) vertical bar C-d vertical bar <= (d - 1) log(m) vertical bar K vertical bar. 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.