On symmetric/asymmetric Lee distance error control codes and elementary symmetric functions

Abstract

This paper gives some new theory and design of codes capable of correcting/detecting errors measured under the Lee distance defined over m-ary words, m ∈ IN. Based on the elementary symmetric functions (instead of the power sums), a key equation is derived which can be used to design new symmetric (or, asymmetric) error control algorithms for some new and already known error control codes for the Lee metric. In particular, it is shown that if K is any field with characteristic char(K) = p, p odd, and u, h, n, m = up <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</sup> , t ∈ IN are such that n ≤ (|K| <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-</sup> 1)/2 and t ≤ (p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</sup> - 1)/2 then there exist m-ary codes C of length n and cardinality |C| ≥ m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> /|K| <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sup> which are capable of, say, correcting t symmetric errors (i. e., the minimum Lee distance of C is d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Lee</sub> (C) ≥ 2t + 1) with t steps of the Extended Euclidean Algorithm. Furthermore, if t ≤ (p - 1)/2 then some of these codes are (essentially) linear and, hence, easy to encode.