On Some New $\mathbb{Z}_{m}$ Linear Codes Based on Elementary Symmetric Functions
Luca G. Tallini, Bella Bose · 2018
Let ℤm=def{0, 1, ... , (m - 1)} be the m-ary alphabet, m ∈ ℕ. This paper gives some new theory and efficient designs of ℤmlinear error control codes based on the elementary symmetric functions of m-ary words. Here, a ℤmlinear code is a sub-module of the module (ℤmn, + mod m, ℤm, · mod m), n ∈ ℕ, and the errors are measured in the L1or Lee metric. In particular, given a field, K, of characteristic p = char(K) = 2, 3, 5, ... prime, and given d, m = vpl, v, l, n ∈ ℕ with d ≤ m/v = pland n ≤ |K|-1, we introduce a new class of (d-1) asymmetric error correcting ℤmlinear codes, Cd, of length n whose redundancy is only ρ(Cd) = n - logm|Cd| ≤ (d - 1) logm|K|. 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.