Variable-Length Codes Independent or Closed with respect to Edit Relations

Abstract

We investigate inference of variable-length codes in other domains of computer science, such as noisy information transmission or information retrieval-storage: in such topics, traditionally mostly constant-length codewords act. The study is relied upon the two concepts of independent and closed sets: given an alphabet A and a binary relationWe focus to those word relations whose images are computed by applying some peculiar combinations of deletion, insertion, or substitution. In particular, characterizations of variable-length codes that are maximal in the families of τ -independent or τ -closed codes are provided.

Gray Cycles of Maximum Length Related to k-Character Substitutions

“…For instance, given the alphabet A = {0, 1}, take for τ the word binary relation Λ 1 which, with every word w associates all the strings located within a Levenshtein distance of 1 from w (see e.g. [17]); with such a relation, the sequence (0, 00, 01, 11, 10, 1) is a…”

“…[20]). Secondly, given a binary word relation τ ⊆ A * × A * , and given X ⊆ A * , if some τ -Gray cycle exists over X, then X is τ -closed [17] that is, the inclusion τ (X) ⊆ X holds, where τ (X) stands for the set of the images of the words in X under the relation τ . Such closed sets actually constitute a special subfamily in the famous dependence systems (see [5] or [12,Sect.…”

“…We focus on the case where τ is σ k , the so-called k-character substitution: with every word with length at least k, say w, this relation associates all the words w ′ , with |w ′ | = |w| and such that the character w ′ i differs from w i in exactly k values of i ∈ [1, |w|]. As commented in [12,17], σ k has noticeable inference in the famous framework of error detection. On the other hand, by definition, w ′ ∈ σ k (w) implies |w ′ | = |w| therefore, if there is some σ k -Gray cycle over X, then X is a uniform set.…”

“…In addition, in the case where A is a binary alphabet, it can be easily proved that, for every n ≥ 3, we have λ A,σ2 (n) = 2 n−1 [14, Exercice 8, p. 77]. However, in the most general case, although an exhaustive description of σ k -closed variable-length codes has been provided in [17], the question of computing some σ k -Gray cycle of maximum length has remained open. In our paper we establish the following result:…”