On Some New $\mathbb{Z}_{m}$ Linear Codes Based on Elementary Symmetric Functions

Let ℤ
_m
=
^def
{0, 1, ... , (m - 1)} be the m-ary alphabet, m ∈ ℕ. 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, ℤ
_m
, · mod m), n ∈ 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, n ∈ ℕ with d ≤ m/v = p
^l
and n ≤ |K|-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 ρ(C
_d
) = n - log
_m
|C
_d
| ≤ (d - 1) log
_m
|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.