Polynomial Invariant Theory and Shape Enumerator of Self-Dual Codes in the NRT-Metric
Welington Santos, Marcelo Muniz S. Alves · IEEE Transactions on Information Theory · 2020
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 Mn,2(F2). 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 Mn,s(F2). We define the ordered flip of a matrix A ∈ Mk,ns(Fq) and present some constructions of self-dual NRT codes over Fq. We further give an application of ordered flip to the classification of self-dual NRT codes of dimension two.