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

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.