Bounds and Genericity of Sum-Rank-Metric Codes

TL;DR

This paper introduces simplified bounds for sum-rank-metric codes, explores their asymptotic behavior, and demonstrates that random linear codes typically approach optimal bounds in this metric.

Contribution

It provides more efficient bounds for sum-rank-metric codes and establishes their asymptotic properties and genericity results for random codes.

Findings

Simplified sphere-packing and Gilbert--Varshamov bounds for sum-rank metric

Asymptotic bounds for growing block size in codes

High probability that random linear codes meet the Gilbert--Varshamov bound

Abstract

We derive simplified sphere-packing and Gilbert--Varshamov bounds for codes in the sum-rank metric, which can be computed more efficiently than previous ones. They give rise to asymptotic bounds that cover the asymptotic setting that has not yet been considered in the literature: families of sum-rank-metric codes whose block size grows in the code length. We also provide two genericity results: we show that random linear codes achieve almost the sum-rank-metric Gilbert--Varshamov bound with high probability. Furthermore, we derive bounds on the probability that a random linear code attains the sum-rank-metric Singleton bound, showing that for large enough extension fields, almost all linear codes achieve it.