Generalized Random Gilbert-Varshamov Codes: Typical Error Exponent and Concentration Properties

TL;DR

This paper determines the exact typical error exponent for generalized random Gilbert-Varshamov codes over DMCs, showing it equals the expurgated error exponent and analyzing concentration properties of the error probabilities.

Contribution

It provides the exact typical error exponent for RGV codes over DMCs and characterizes the concentration rates and tail decay behaviors.

Findings

Typical error exponent equals the expurgated error exponent.

Convergence of the random coding exponent to the typical error exponent.

Exponential decay of the lower tail and double exponential decay of the upper tail.

Abstract

We find the exact typical error exponent of constant composition generalized random Gilbert-Varshamov (RGV) codes over DMCs channels with generalized likelihood decoding. We show that the typical error exponent of the RGV ensemble is equal to the expurgated error exponent, provided that the RGV codebook parameters are chosen appropriately. We also prove that the random coding exponent converges in probability to the typical error exponent, and the corresponding non-asymptotic concentration rates are derived. Our results show that the decay rate of the lower tail is exponential while that of the upper tail is double exponential above the expurgated error exponent. The explicit dependence of the decay rates on the RGV distance functions is characterized.