Cumulative Distance Enumerators of Random Codes and their Thresholds

TL;DR

This paper introduces cumulative weight enumerators for random linear codes, studies their asymptotic behavior, and identifies sharp thresholds for code properties related to relative distance, connecting to the Gilbert-Varshamov bound.

Contribution

It establishes the asymptotic properties and sharp threshold points for the density of codes exceeding certain relative distances, extending to arbitrary random codes.

Findings

Sharp threshold at the Gilbert-Varshamov bound for linear codes.

Threshold at half the Gilbert-Varshamov bound for arbitrary codes.

Asymptotic analysis of cumulative weight enumerators.

Abstract

Cumulative weight enumerators of random linear codes are introduced, their asymptotic properties are studied, and very sharp thresholds are exhibited; as a consequence, it is shown that the asymptotic Gilbert-Varshamov bound is a very sharp threshold point for the density of the linear codes whose relative distance is greater than a given positive number. For arbitrary random codes, similar settings and results are exhibited; in particular, the very sharp threshold point for the density of the codes whose relative distance is greater than a given positive number is located at half the asymptotic Gilbert-Varshamov bound.