Theory of supports for linear codes endowed with the sum-rank metric

TL;DR

This paper develops a theoretical framework for supports in sum-rank metric codes, extending classical concepts and establishing duality, bounds, and properties relevant for error correction in multishot matrix channels.

Contribution

It introduces sum-rank supports, explores their lattice structure, and extends classical code operations, providing foundational results and applications for sum-rank metric coding theory.

Findings

Sum-rank supports form a lattice structure.

Duality and support spaces are characterized.

Bounds and properties for restricted and shortened codes are established.

Abstract

The sum-rank metric naturally extends both the Hamming and rank metrics in coding theory over fields. It measures the error-correcting capability of codes in multishot matrix-multiplicative channels (e.g. linear network coding or the discrete memoryless channel on fields). Although this metric has already shown to be of interest in several applications, not much is known about it. In this work, sum-rank supports for codewords and linear codes are introduced and studied, with emphasis on duality. The lattice structure of sum-rank supports is given; characterizations of the ambient spaces (\textit{support spaces}) they define are obtained; the classical operations of restriction and shortening are extended to the sum-rank metric; and estimates (bounds and equalities) on the parameters of such restricted and shortened codes are found. Three main applications are given: 1) Generalized sum-rank weights are introduced, together with their basic properties and bounds; 2) It is shown that duals, shortened and restricted codes of maximum sum-rank distance (MSRD) codes are in turn MSRD; 3) Degenerateness and effective lengths of sum-rank codes are introduced and characterized. In an appendix, skew supports are introduced, defined by skew polynomials, and their connection to sum-rank supports is given.