Rate-distance tradeoff for codes above graph capacity

TL;DR

This paper investigates the maximum rate of zero-undetected-error communication over graphs with adversarial noise, introducing bounds based on graph homomorphisms, the Gilbert-Varshamov bound, and Delsarte's linear programming method.

Contribution

It generalizes the graph capacity concept by incorporating a fraction of differing coordinates and derives new lower and upper bounds using advanced graph-theoretic techniques.

Findings

Derived lower bounds using graph homomorphisms and Gilbert-Varshamov bound.

Established upper bounds via Delsarte's linear programming and Lovász' theta function.

Provided a framework for understanding zero-error communication with adversarial noise.

Abstract

The capacity of a graph is defined as the rate of exponential growth of independent sets in the strong powers of the graph. In the strong power an edge connects two sequences if at each position their letters are equal or adjacent. We consider a variation of the problem where edges in the power graphs are removed between sequences which differ in more than a fraction $\delta$ of coordinates. The proposed generalization can be interpreted as the problem of determining the highest rate of zero undetected-error communication over a link with adversarial noise, where only a fraction $\delta$ of symbols can be perturbed and only some substitutions are allowed. We derive lower bounds on achievable rates by combining graph homomorphisms with a graph-theoretic generalization of the Gilbert-Varshamov bound. We then give an upper bound, based on Delsarte's linear programming approach, which combines Lov\'asz' theta function with the construction used by McEliece et al. for bounding the minimum distance of codes in Hamming spaces.