Various Views on the Trapdoor Channel and an Upper Bound on its Capacity

TL;DR

This paper introduces two new perspectives on the trapdoor channel, modeling it as a fractal through iterated function systems and providing an algorithm for its characterization, ultimately establishing a new upper bound on its capacity.

Contribution

It presents novel fractal-based views and an algorithm for the trapdoor channel, and derives the tightest known upper bound on its capacity, showing feedback increases capacity.

Findings

Fractal approximation of the trapdoor channel via IFS

An algorithm characterizing the trapdoor channel's permutations

Upper bound of approximately 0.6610 bits per use on capacity

Abstract

Two novel views are presented on the trapdoor channel. First, by deriving the underlying iterated function system (IFS), it is shown that the trapdoor channel with input blocks of length $n$ can be regarded as the $n$th element of a sequence of shapes approximating a fractal. Second, an algorithm is presented that fully characterizes the trapdoor channel and resembles the recursion of generating all permutations of a given string. Subsequently, the problem of maximizing a $n$-letter mutual information is considered. It is shown that $\frac{1}{2}\log_2\left(\frac{5}{2}\right)\approx 0.6610$ bits per use is an upper bound on the capacity of the trapdoor channel. This upper bound, which is the tightest upper bound known proves that feedback increases capacity of the trapdoor channel.