Densities of Codes of Various Linearity Degrees in Translation-Invariant Metric Spaces

TL;DR

This paper studies how the density of error-correcting codes with various linearity degrees behaves asymptotically in different translation-invariant metric spaces, revealing dependence on linearity and metric choice.

Contribution

It provides a comprehensive analysis of the asymptotic density of codes with different linearity degrees across various translation-invariant metrics, including Hamming, rank, and sum-rank.

Findings

Density varies significantly with linearity degree and metric.

Results apply to finite translation-invariant metric spaces.

Specialized results for Hamming, rank, and sum-rank metrics.

Abstract

We investigate the asymptotic density of error-correcting codes with good distance properties and prescribed linearity degree, including sublinear and nonlinear codes. We focus on the general setting of finite translation-invariant metric spaces, and then specialize our results to the Hamming metric, to the rank metric, and to the sum-rank metric. Our results show that the asymptotic density of codes heavily depends on the imposed linearity degree and the chosen metric.