Markov Chain + Decoding algorithm

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I am ready a paper Canteaut and Chabaud, I don't get understand the values of transition matrix $P$, in the Proposition 4. If, anybody read this paper please help me understand this values: $P_{u,u}$, $P_{u,u-1}$, etc.

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$P_{u,v}$ is the probability of a transition between state $u$ to state $v$. It's described more formally in Definition 3 where for a given Markov chain $X$, the conditional probability $P(X_i = v | X_{i-1} = u) = P_{u,v}$.