Bounds on the Reliability of a Typewriter Channel

TL;DR

This paper establishes new theoretical bounds on the reliability function of a specific typewriter channel with 5 inputs and a crossover probability of 1/2, challenging previous conjectures and refining existing bounds.

Contribution

It provides the first counterexample to a conjecture on the tightness of the expurgated bound without relying on algebraic-geometric codes.

Findings

The lower bound marginally improves the expurgated bound.

The upper bound is derived via an adapted linear programming bound.

The results serve as a low-rate anchor for the straight line bound.

Abstract

We give new bounds on the reliability function of a typewriter channel with 5 inputs and crossover probability $1/2$. The lower bound is more of theoretical than practical importance; it improves very marginally the expurgated bound, providing a counterexample to a conjecture on its tightness by Shannon, Gallager and Berlekamp which does not need the construction of algebraic-geometric codes previously used by Katsman, Tsfasman and Vl\u{a}du\c{t}. The upper bound is derived by using an adaptation of the linear programming bound and it is essentially useful as a low-rate anchor for the straight line bound.