Bounds on the reliability of typewriter channels

Abstract

New lower and upper bounds on the reliability function of typewriter channels are given. Our lower bounds improve upon the (multiletter) expurgated bound of Gallager, furnishing a new and simple counterexample to a conjecture made in 1967 by Shannon, Gallager and Berlekamp on its tightness. The only other known counterexample is due to Katsman, Tsfasman and Vl\u{a}du\c{t} who used algebraic-geometric codes on a $q$-ary symmetric channels, $q\geq 49$. Here we prove, by introducing dependence between codewords of a random ensemble, that the conjecture is false even for a typewriter channel with $q=4$ inputs. In the process, we also demonstrate that Lov\'asz's proof of the capacity of the pentagon was implicitly contained (but unnoticed!) in the works of Jelinek and Gallager on the expurgated bound done at least ten years before Lov\'asz. In the opposite direction, new upper bounds on the reliability function are derived for channels with an odd number of inputs by using an adaptation of Delsarte's linear programming bound. First we derive a bound based on the minimum distance, which combines Lov\'asz's construction for bounding the graph capacity with the McEliece-Rodemich-Rumsey-Welch construction for bounding the minimum distance of codes in the Hamming space. Then, for the particular case of cross-over probability $1/2$, we derive an improved bound by also using the method of Kalai and Linial to study the spectrum distribution of codes.