Fundamental Properties of Sum-Rank Metric Codes

TL;DR

This paper develops the theory of sum-rank metric codes with variable block sizes, establishing bounds, duality properties, and constructions, advancing understanding of their structure and limitations.

Contribution

It introduces new bounds, explores duality conditions, and provides constructions for sum-rank metric codes with variable block sizes.

Findings

MSRD codes dualize only when all blocks have the same number of columns

Bounds on code cardinality are derived and extended asymptotically

Constructs examples illustrating bounds, existence, and duality of sum-rank codes

Abstract

This paper investigates the theory of sum-rank metric codes for which the individual matrix blocks may have different sizes. Various bounds on the cardinality of a code are derived, along with their asymptotic extensions. The duality theory of sum-rank metric codes is also explored, showing that MSRD codes (the sum-rank analogue of MDS codes) dualize to MSRD codes only if all matrix blocks have the same number of columns. In the latter case, duality considerations lead to an upper bound on the number of blocks for MSRD codes. The paper also contains various constructions of sum-rank metric codes for variable block sizes, illustrating the possible behaviours of these objects with respect to bounds, existence, and duality properties.