Elias Bound for General Distances and Stable Sets in Edge-Weighted Graphs

TL;DR

This paper extends the Elias bound to general distances and connects it with Lovász's graph capacity bound, unifying multiple bounds and applying to code reliability analysis.

Contribution

It introduces a unified bound that generalizes Elias, Lovász, and previous bounds, applicable to codes with arbitrary distances and graph capacities.

Findings

Unified bound encompasses Elias and Lovász bounds

Includes previous bounds as special cases

Applicable to reliability function analysis

Abstract

This paper presents an extension of the Elias bound on the minimum distance of codes for discrete alphabets with general, possibly infinite-valued, distances. The bound is obtained by combining a previous extension of the Elias bound, introduced by Blahut, with an extension of a bound previously introduced by the author which builds upon ideas of Gallager, Lov\'asz and Marton. The result can in fact be interpreted as a unification of the Elias bound and of Lov\'asz's bound on graph (or zero-error) capacity, both being recovered as particular cases of the one presented here. Previous extensions of the Elias bound by Berlekamp, Blahut and Piret are shown to be included as particular cases of our bound. Applications to the reliability function are then discussed.