The Typical Non-Linear Code over Large Alphabets

TL;DR

This paper investigates the typical properties of non-linear codes over large alphabets, revealing they often do not achieve maximum distance, and provides asymptotic analysis for code distance properties in both block and subspace codes.

Contribution

It characterizes the typical non-linear codes over large alphabets, contrasting their behavior with linear codes, and extends results to subspace codes with applications in finite geometry.

Findings

Typical non-linear codes are far from being MDS in the Hamming metric.

Asymptotic proportion of codes with good distance properties is characterized.

Results extend to subspace codes, with applications to finite geometry.

Abstract

We consider the problem of describing the typical (possibly) non-linear code of minimum distance bounded from below over a large alphabet. We concentrate on block codes with the Hamming metric and on subspace codes with the injection metric. In sharp contrast with the behavior of linear block codes, we show that the typical non-linear code in the Hamming metric of cardinality $q^{n-d+1}$ is far from having minimum distance $d$, i.e., from being MDS. We also give more precise results about the asymptotic proportion of block codes with good distance properties within the set of codes having a certain cardinality. We then establish the analogous results for subspace codes with the injection metric, showing also an application to the theory of partial spreads in finite geometry.