Communicating over the Torn-Paper Channel

TL;DR

This paper analyzes the capacity of a novel torn-paper communication channel where messages are randomly fragmented and shuffled, revealing that random fragment lengths can lead to higher capacity than deterministic ones due to occasional large fragments.

Contribution

It provides a precise capacity characterization for the binary torn-paper channel with geometric fragment lengths and highlights the qualitative difference from deterministic fragment lengths.

Findings

Capacity is characterized as C = e^{-eta} with  = _{n o \u221e} p_n \, log n.

Random geometric fragment lengths can yield higher capacity than fixed lengths.

Large fragments in the random case boost the overall channel capacity.

Abstract

We consider the problem of communicating over a channel that randomly "tears" the message block into small pieces of different sizes and shuffles them. For the binary torn-paper channel with block length $n$ and pieces of length ${\rm Geometric}(p_n)$, we characterize the capacity as $C = e^{-\alpha}$, where $\alpha = \lim_{n\to\infty} p_n \log n$. Our results show that the case of ${\rm Geometric}(p_n)$-length fragments and the case of deterministic length-$(1/p_n)$ fragments are qualitatively different and, surprisingly, the capacity of the former is larger. Intuitively, this is due to the fact that, in the random fragments case, large fragments are sometimes observed, which boosts the capacity.