The Capacities of Periodic-Finite-Type Shifts from the Perspective of the Forbidden-Word Length

抄録

A Periodic-Finite-Type shift (PFT) is a set of bi-infinite sequences that prohibit the appearance of forbidden words in a periodic manner. More precisely, a PFT $\cX_{\{T, \tilde \F\}}$ is a set of bi-infinite sequences $\x$ characterized by a period $T\in \N$ and a family $\tilde\F=(\tilde\cF^{(0)},\tilde\cF^{(1)}, \cdots, \tilde\cF^{(T-1)})$ of indexed finite sets of forbidden words $\tilde\cF^{(0)},\tilde\cF^{(1)}, \cdots, \tilde\cF^{(T-1)}$, so that the $r$-shifted sequence $\sigma^r(\x)$ of $\x$ does not contain words in $\tilde\cF^{(i \mod T)}$ at position $i\in \Z$. The study on PFTs is strongly related to the study on constrained systems with unconstrained positions, which have the property as both error-correcting codes and constrained codes.

The capacity of a PFT is an important value that gives us the maximum coding rate when a random sequence is encoded to a sequence in the PFT. In this paper, we derive the capacity of a PFT in two ways, using the fact that an arbitrary family $\tilde \F$ is transformed into a family $\F=(\cF^{(0)},\emptyset \cdots, \emptyset)$, where each forbidden word in $\cF^{(0)}$ has the same length $k$, so that $\cX_{\{T, \tilde \F\}}=\cX_{\{T, \F\}}$. When $k \le T$, the first proof derives the capacity directly from the definition, and the other proof does from block partitioning of the adjacency matrix of a certain graph representing $\cX_{\{T, \F\}}$. We also present a partial result on the capacity when $k>T$.