Invariants for sum-rank metric codes

TL;DR

This paper introduces new invariants and a computational framework for sum-rank metric codes, enabling more effective analysis of code equivalence where traditional methods are insufficient.

Contribution

It develops novel invariants and nuclear parameters for sum-rank codes, along with a skew polynomial-based computational approach, advancing the study of code equivalence.

Findings

New invariants for sum-rank codes are introduced.

Nuclear parameters are defined and used to distinguish inequivalent codes.

A computational framework using skew polynomials is developed for explicit calculations.

Abstract

The code equivalence problem is central in coding theory and cryptography. While classical invariants are effective for Hamming and rank metrics, the sum-rank metric, which unifies both, introduces new challenges. This paper introduces new invariants for sum-rank metric codes: generalised idealisers, the centraliser, the center, and a refined notion of linearity. These lead to the definition of nuclear parameters, inspired by those used in division algebra theory, where they are crucial for proving inequivalence. We also develop a computational framework based on skew polynomials, which is isometric to the classical matrix setting but enables explicit computation of nuclear parameters for known MSRD (Maximum Sum-Rank Distance) codes. This yields a new and effective method to study the code equivalence problem where traditional tools fall short. In fact, using nuclear parameters, we can study the equivalence among the largest families of known MSRD codes.