Algebraic dynamical systems from LDPC codes satisfy a strong negation of the weak Pinsker property

TL;DR

This paper constructs an algebraic dynamical system derived from LDPC codes that exhibits high sofic entropy but lacks a Bernoulli factor, revealing complex clustering behavior in its model spaces.

Contribution

It provides the first explicit algebraic example of a subshift with positive sofic entropy that does not admit a Bernoulli factor, using LDPC code structures.

Findings

The constructed system has completely positive sofic entropy.

Its model spaces break into exponentially many small clusters.

The example links algebraic dynamics with coding theory insights.

Abstract

We construct an explicit algebraic example of a subshift of finite type over a group $\Gamma$ with an invariant Markov measure which has completely positive sofic entropy (with respect to `most' sofic approximations) and yet does not have a direct Bernoulli factor, because its model spaces shatter into exponentially many clusters of sub-exponential size. The example and its analysis are related to random low-density parity-check (LDPC) codes.