Generalized Random Gilbert-Varshamov Codes

TL;DR

This paper introduces a new random coding technique inspired by the Gilbert-Varshamov bound, providing tight error exponents and optimal code constructions for discrete memoryless channels.

Contribution

It presents a novel recursive code construction based on type classes and minimum distance, extending to infinite alphabets and general decoding metrics.

Findings

Achieves a tight error exponent matching the ensemble average.

Recovers the Csiszár-Körner exponent as a special case.

Establishes the optimality of certain distance functions.

Abstract

We introduce a random coding technique for transmission over discrete memoryless channels, reminiscent of the basic construction attaining the Gilbert-Varshamov bound for codes in Hamming spaces. The code construction is based on drawing codewords recursively from a fixed type class, in such a way that a newly generated codeword must be at a certain minimum distance from all previously chosen codewords, according to some generic distance function. We derive an achievable error exponent for this construction, and prove its tightness with respect to the ensemble average. We show that the exponent recovers the Csisz\'{a}r and K{\"o}rner exponent as a special case, which is known to be at least as high as both the random-coding and expurgated exponents, and we establish the optimality of certain choices of the distance function. In addition, for additive distances and decoding metrics, we present an equivalent dual expression, along with a generalization to infinite alphabets via cost-constrained random coding.