Sofic measures

Sofic measures, also known as hidden Markov measures, have been extensively studied. In this paper, we survey some equivalent definitions of this notion and improve a bound for deciding whether a sofic measure is~$k$-step Markov. We prove that if an invariant sofic measure with a linear representation of dimension~$n$ is a~$k$-step Markov chain, then~$k$ can be chosen at most equal to~$2^{n^2-1}$.