SOME PROPERTIES OF MAXIMAL COMMA-FREE CODES

Abstract

A code $A$ over a finite alphabet $X$ is a comma-free code if $A^2\cap X^+AX^+=\emptyset$, where $X$ is a finite alphabet containing more than one letter. This paper is a study of some algebraic properties of finite maximal comma-free codes. We give several characterizations on two-element comma-free codes and maximal comma-free codes. Let $X_n^m= X^n\cup X^{n+1}\cup \cdots \cup X^m$. We prove that for $n \ge 3$ , a maximal comma-free code in $X^n$ is a maximal comma-free code in the region $X_1^m\cup X^n$, $m < n/2$. We also obtain that for $X = \{a,b\}$, a maximal comma-free code in $X^3$ is a maximal comma-free code; a maximal comma-free code in $X^4$ is a maximal comma-free code in $X_1^4$; for every $n\ge 4$, there is a maximal comma-free code in $X^n$ which is not a maximal comma-free code.