Competitive minimax universal decoding for several ensembles of random codes

TL;DR

This paper develops a universal decoding approach that guarantees near-optimal error exponents across various code ensembles and channels, demonstrating its effectiveness especially for BSC and linear codes.

Contribution

It introduces a single-letter expression for the maximum achievable fraction of the optimal error exponent universally, and proves its tightness for several important code ensembles.

Findings

Universal decoding achieves the full error exponent for BSC with uniform code ensembles.

The proposed method is efficiently implementable using modified Viterbi algorithms.

The approach applies to systematic linear and time-varying convolutional codes.

Abstract

Universally achievable error exponents pertaining to certain families of channels (most notably, discrete memoryless channels (DMC's)), and various ensembles of random codes, are studied by combining the competitive minimax approach, proposed by Feder and Merhav, with Chernoff bound and Gallager's techniques for the analysis of error exponents. In particular, we derive a single--letter expression for the largest, universally achievable fraction $\xi$ of the optimum error exponent pertaining to the optimum ML decoding. Moreover, a simpler single--letter expression for a lower bound to $\xi$ is presented. To demonstrate the tightness of this lower bound, we use it to show that $\xi=1$, for the binary symmetric channel (BSC), when the random coding distribution is uniform over: (i) all codes (of a given rate), and (ii) all linear codes, in agreement with well--known results. We also show that $\xi=1$ for the uniform ensemble of systematic linear codes, and for that of time--varying convolutional codes in the bit-error--rate sense. For the latter case, we also show how the corresponding universal decoder can be efficiently implemented using a slightly modified version of the Viterbi algorithm which em employs two trellises.