Polynomial Invariant Theory and Shape Enumerator of Self-Dual Codes in the NRT-Metric

In this paper we consider self-dual NRT codes, that is, self-dual codes in the metric space endowed with the Niederreiter-Rosenbloom-Tsfasman metric (NRT metric) and their shape enumerators as defined by Barg and Park. We use polynomial invariant theory to describe the shape enumerator of a binary self-dual NRT code, even self-dual NRT code, and weak doubly even self-dual NRT code in M
_n,2
(F
₂
). Motivated by these results, we describe the number of invariant polynomials that we must find to describe the shape enumerator of a self-dual NRT code in M
_n,s
(F
₂
). We define the ordered flip of a matrix A ∈ M
_k,ns
(F
_q
) and present some constructions of self-dual NRT codes over F
_q
. We further give an application of ordered flip to the classification of self-dual NRT codes of dimension two.