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We construct an explicit algebraic example of a subshift of finite type overa group $\Gamma$ with an invariant Markov measure which has completely positivesofic entropy (with respect to `most' sofic approximations) and yet does nothave a direct Bernoulli factor, because its model spaces shatter intoexponentially many clusters of sub-exponential size. The example and itsanalysis are related to random low-density parity-check (LDPC) codes.

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