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}(\mathbb {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}}(\mathbb {F}_{2})$. We define the ordered flip of a matrix $ {A}\in {M}_{ {k},{ { ns}}}(\mathbb {F}_{ {q}})$ and present some constructions of self-dual NRT codes over $\mathbb {F}_{ {q}}$. We further give an application of ordered flip to the classification of self-dual NRT codes of dimension two. [ABSTRACT FROM AUTHOR]