An "Umbrella" Bound of the Lov\'asz-Gallager Type

TL;DR

This paper introduces a new bound on the error probability of discrete memoryless channels by combining Lovász and Gallager ideas, using a novel function that bridges known channel parameters.

Contribution

It proposes a new function $ heta( ho)$ that interpolates between the cut-off rate and Lovász theta, providing a unified bound on error probability.

Findings

Bound is finite for all rates above Lovász theta

Introduces the $ heta( ho)$ function linking key channel parameters

Bound, though loose, extends error analysis to broader rate regions

Abstract

We propose a novel approach for bounding the probability of error of discrete memoryless channels with a zero-error capacity based on a combination of Lov\'asz' and Gallager's ideas. The obtained bounds are expressed in terms of a function $\vartheta(\rho)$, introduced here, that varies from the cut-off rate of the channel to the Lov\'azs theta function as $\rho$ varies from 1 to $\infty$ and which is intimately related to Gallager's expurgated coefficient. The obtained bound to the reliability function, though loose in its present form, is finite for all rates larger than the Lov\'asz theta function.