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

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.