Explicit Constructions of Sum-Rank Metric Codes from Quadratic Kummer Extensions

TL;DR

This paper presents new algebraic constructions of sum-rank metric codes using quadratic Galois extensions, achieving larger block lengths and explicit parameters, with potential for improved code performance.

Contribution

It introduces two novel constructions of sum-rank codes from quadratic Kummer extensions, expanding the known code parameters and analyzing their asymptotic properties.

Findings

Codes achieve larger block lengths than previous constructions

Explicit parameters including dimensions and minimum distances are determined

Codes exhibit favorable asymptotic behavior compared to bounds

Abstract

This paper introduces new constructions of sum-rank metric codes derived from algebraic function fields, as existing results on such codes remain limited. A major challenge lies in the determination of their parameters. We address this issue by employing quadratic Galois extensions, proposing two general constructions of $2\times2$ sum-rank codes. Analogous to algebraic geometry codes in the Hamming metric, our codes achieve a larger block length compared to existing constructions. We determine explicit parameters including dimensions and minimum distances of our codes, and we present an illustrative example using elliptic function fields. Finally, we discuss the asymptotic behavior of our codes and compare them with the Gilbert-Varshamov-like bound for sum-rank metric codes.