Maximal Comma-free Codes Research Articles

Hall sets, Lazard sets and comma-free codes

We investigate the relationship between two constructions of maximal comma-free codes described, respectively, by Eastman and by Scholtz and the notions of Hall sets and Lazard sets introduced in connection with factorizations of free monoids and bases of free Lie algebras.

Codon Distribution in Error-Detecting Circular Codes.

In 1957, Francis Crick et al. suggested an ingenious explanation for the process of frame maintenance. The idea was based on the notion of comma-free codes. Although Crick’s hypothesis proved to be wrong, in 1996, Arquès and Michel discovered the existence of a weaker version of such codes in eukaryote and prokaryote genomes, namely the so-called circular codes. Since then, circular code theory has invariably evoked great interest and made significant progress. In this article, the codon distributions in maximal comma-free, maximal self-complementary and maximal self-complementary circular codes are discussed, i.e., we investigate in how many of such codes a given codon participates. As the main (and surprising) result, it is shown that the codons can be separated into very few classes (three, or five, or six) with respect to their frequency. Moreover, the distribution classes can be hierarchically ordered as refinements from maximal comma-free codes via maximal self-complementary codes to maximal self-complementary circular codes.

Varieties of comma-free codes

New varieties of comma-free codes CFC of length 3 on the 4-letter alphabet are defined and analysed: self-complementary comma-free codes (CCFC), C3 comma-free codes (C3CFC), C3 self-complementary comma-free codes (C3CCFC), self-complementary maximal comma-free codes (CMCFC), C3 maximal comma-free codes (C3MCFC) and C3 self-complementary maximal comma-free codes (C3CMCFC). New properties with words of length 3, 4, 5 and 6 in comma-free codes are used for the determination of growth functions in the studied code varieties.

Finite Completion of comma-free codes Part 2

This paper is a sequel to an earlier paper of the present author, in which it was proved that every finite comma-free code is embedded into a so-called (finite) canonical comma-free code. In this paper, it is proved that every (finite) canonical comma-free code is embedded into a finite maximal comma-free code, which thus achieves the conclusion that every finite comma-free code has finite completions.

Finite Completion of comma-free codes Part 1

This paper is the first step in the solution of the problem of finite completion of comma-free codes. We show that every finite comma-free code is included in a finite comma-free code of particular kind, which we called, for lack of a better term, canonical comma-free code. Certainly, finite maximal comma-free codes are always canonical. The final step of the solution which consists in proving further that every canonical comma-free code is completed to a finite maximal comma-free code, is intended to be published in a forthcoming paper.

Completing comma-free codes

We prove that for every regular comma-free code there exists a maximal comma-free code containing it which is still regular and, moreover, we can effectively locate such a completion. In particular, we can decide whether a given regular comma-free code is maximal.

SOME PROPERTIES OF MAXIMAL COMMA-FREE CODES

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.

Finite State Models in the Study of Comma-Free Codes

The definition of comma-free codes is extended to be more comprehensive and include codes with variable length codewords and codes with an unlimited number of codewords. Finite state recognizers are used to represent commafree codes in a compact form, making available the well established methods of manipulating and analysing finite state models. An efficient algorithm is presented to test the comma-freeness of any given code represented by a finite state recognizer. It is well known, that any maximal comma-free code with codewords of odd fixed length can be achieved. Although upper limits have been given for even fixed length it has not been established which comma-free codes of this nature can achieve the upper limit. Compact representations of comma-free binary codes are given for variable length codewords with maximum even fixed lengths (6, 8 and 10) and in one case (length not greater than 6) it is seen that the corresponding upper limit for fixed lengths can be exceeded.

The comma-free codes with words of length two

A code not requiring a distinct symbol to separate words is called comma-free. Two codes are isomorphic if one can be obtained from the other by a permutation of the underlying alphabet. Since subcodes of comma-free codes are comma-free, we investigate only maximal comma-free codes. All isomorphism classes of maximal comma-free codes with words of length 2 are determined and a natural representative of each class is given.

On maximal comma-free codes (Corresp.)

In this correspondence a binary maximal comma-free code for words of length 10 and the search technique used to construct it are described.