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

## TL;DR

This paper applies polynomial invariant theory to characterize the shape enumerator of self-dual NRT-codes, providing new insights into their invariants, classifications, and constructions in the context of the NRT-metric.

## Contribution

It introduces a method to describe shape enumerators of self-dual NRT-codes using invariant polynomials and explores the classification and construction of such codes.

## Key findings

- Derived formulas for shape enumerators of self-dual NRT-codes.
- Count of invariant polynomials needed for classification.
- New constructions of self-dual NRT-codes over finite fields.

## Abstract

In this paper we consider self-dual NRT-codes, that is, self-dual codes in the metric space endowed with the Niederreiter-Rosenbloom-Tsfasman (NRT-metric). We use polynomial invariant theory to describe the shape enumerator of a binary self-dual, doubly even self-dual, and doubly-doubly even self dual NRT-code $C\subseteq M_{n,2}(\mathbb{F}_{2})$. Motivated by these results we describe the number of invariant polinomials that we must find to describe the shape enumerator of a self-dual NRT-code of $M_{n,s}(\mathbb{F}_{2})$. 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 bidimensional self-dual NRT-codes.