Topologies for Error-Detecting Variable-Length Codes

Given a finite alphabet $A$, a quasi-metric $d$ over $A^*$, and a non-negative integer $k$, we introduce the relation $\tau_{d,k}\subseteq A^*\times A^*$ such that $(x,y)\in\tau_{d,k}$ holds whenever $d(x,y)\le k$. The error detection capability of variable-length codes is expressed in term of conditions over $\tau_{d,k}$. With respect to the prefix metric, the factor one, and any quasi-metric associated with some free monoid (anti-)automorphism, we prove that one can decide whether a given regular variable-length code satisfies any of those error detection constraints.Comment: arXiv admin note: text overlap with arXiv:2208.1468

Finite maximal solid codes

AbstractSolid codes, a special class of bifix codes, were introduced recently in the connection with formal languages. However, they have a much earlier history and more important motivation in information transmission dating back to the 1960s. In this paper, they are studied as an independent subject in the theory of variable-length codes. It is shown that every finite solid code is contained in a finite maximal one; based on further analysis of the structure of finite maximal solid codes, an algorithm is proposed to construct all of them starting from the most simple and evident ones