We study recoverable systems on the 2D lattice using tools from statistical mechanics. For a particular recovery rule that we consider as an example, we compute lower and upper bounds on the topological entropy of the system (the case of zero temperature). We also show that at low positive temperature, typical configurations are local perturbations of the ground states. For the case of high temperature, we show uniqueness of the Gibbs measure and find the mixing rate of Glauber dynamics for the system in a finite domain.

Non-overlapping codes are a set of codewords in$\bigcup _{n \ge 2} \mathbb {Z}_{q}^{n}$, where$\mathbb {Z}_{q} = \{0,1, {\dots },q-1\}$, such that the prefix of each codeword is not a suffix of any codeword in the set, including itself; and for variable-length codes, a codeword does not contain any other codeword as a subword. In this paper, we investigate a generic method to generalize binary codes to$q$-ary ones for$q > 2$, and analyze this generalization on the two constructions given by Levenshtein (also by Gilbert; Chee, Kiah, Purkayastha, and Wang) and Bilotta, respectively. The generalization on the former construction gives large non-expandable fixed-length non-overlapping codes whose size can be explicitly determined; the generalization on the latter construction is the first attempt to generate$q$-ary variable-length non-overlapping codes. More importantly, this generic method allows us to utilize the generating function approach to analyze the cardinality of the underlying$q$-ary non-overlapping codes. The generating function approach not only enables us to derive new results, e.g., recurrence relations on their cardinalities, new combinatorial interpretations for the constructions, and the superior limit of their cardinalities for some special cases, but also greatly simplifies the arguments for these results. Furthermore, we give an exact formula for the number of fixed-length words that do not contain the codewords in a variable-length non-overlapping code as subwords. This thereby solves an open problem by Bilotta and induces a recursive upper bound on the maximum size of variable-length non-overlapping codes.