Bifix Codes Research Articles

Some Results on $${(\varvec{k,m})}$$-Comma Codes

Suppose that L is a nonempty language over A, $$k\in \mathbb {N}^0$$ and $$m\in \mathbb {N}$$ . If L satisfies $$(LA^k)^mL$$ $$\cap A^+(LA^k)^{m-1}LA^+=\emptyset $$ , then L is a code and is called a (k, m)-comma code. If L is a (k, m)-comma code, then it is known that L is an infix code when $$m=1$$ and L is a bifix code when $$m\ge 1$$ . In this paper, we extend the study and find that the class of (k, m)-comma codes and the class of infix codes are incomparable but they are not disjoint, so we first characterize the (k, m)-comma codes with $$m\ge 2$$ which are infix codes. We also describe the solid codes with minimum lengths greater than k and the 2-(k, 1)-comma codes by different approaches. Finally, the class of bifix codes will be classified into disjoint union of some subclasses and a criterion to determine the subclass membership of a given bifix code will be given.

Specular sets

We introduce specular sets. These are subsets of groups which form a natural generalization of free groups. These sets of words are an abstract generalization of the natural codings of interval exchanges and of linear involutions. We consider two important families of sets contained in specular sets: sets of return words and maximal bifix codes. For both families we prove several cardinality results as well as results concerning the subgroup generated by these sets.

Maximal Bifix Codes of Degree 3

The construction of thin maximal bifix codes of degree 1 or 2 is clear and simple. In this paper, a class of thin maximal bifix codes of degree 3 which contains all finite maximal bifix codes of degree 3 is investigated. The construction of such codes is determined. It is well known that for any positive integers d and k, there are finitely many finite maximal bifix codes of degree d over a k-letter alphabet. But the enumeration problem is unsolved in general. In this paper, we show that for any positive integer k, there is a bijection from the set of all finite maximal bifix codes of degree 3 over a k-letter alphabet onto the set of all directed acyclic graphs with k vertices, and then, the enumeration problem of finite maximal bifix codes of degree 3 is solved.

Some Properties of Extractable Codes and Insertable Codes

This paper deals with insertability and mainly extractablity of codes. A code C is called insertable (or extractable) if the free submonoid C* generated by C satisfies if z, [Formula: see text] implies [Formula: see text] (or z, [Formula: see text] implies [Formula: see text]). We show that a finite insertable code is a full uniform code. On the other hand there are many finite extractable codes which are not full uniform codes. We cannot still characterize the structures of infinite extractable codes. Here we give some results on the class of infix extractable codes. First, we consider a necessary and sufficient condition whether a given infix code C is extractable or not by using the syntactic graph of the code. Secondly, we investigate the extractability for the families of other related bifix codes. We newly define the bifix codes, called e(m)-codes and [Formula: see text]-codes, and refer to the extractability of them.

Neutral and tree sets of arbitrary characteristic

We study classes of minimal sets defined by restrictions on the possible extensions of the words. These sets generalize the previously studied classes of neutral and tree sets by relaxing the condition imposed on the empty word and measured by an integer called the characteristic of the set. We present several enumeration results holding in these sets of words. These formulae concern return words and bifix codes. They generalize formulae previously known for Sturmian sets or more generally for tree sets. We also give two geometric examples of this class of sets, namely the natural coding of some interval exchange transformations and the natural coding of some linear involutions.

Characterizations of $$k$$ k -comma codes and $$k$$ k -comma intercodes

The notions of $$k$$k-comma codes and $$k$$k-comma intercodes are proper generalizations of comma-free codes and intercodes respectively. In this paper, we characterize $$k$$k-comma codes and $$k$$k-comma intercodes by using infix codes and bifix codes. Then, we consider the homomorphism which preserves $$k$$k-comma codes and $$k$$k-comma intercodes.

The finite index basis property

We describe in this paper a connection between bifix codes, symbolic dynamical systems and free groups. This is in the spirit of the connection established previously for the symbolic systems corresponding to Sturmian words. We introduce a class of sets of factors of an infinite word with linear factor complexity containing Sturmian sets and regular interval exchange sets, namely the class of tree sets. We prove as a main result that for a uniformly recurrent tree set S, a finite bifix code X on the alphabet A is S-maximal of S-degree d if and only if it is the basis of a subgroup of index d of the free group on A.

Bifix codes and interval exchanges

We investigate the relation between bifix codes and interval exchange transformations. We prove that the class of natural codings of regular interval exchange transformations is closed under maximal bifix decoding.

COMPLETELY REDUCIBLE SETS

We study the family of rational sets of words, called completely reducible and which are such that the syntactic representation of their characteristic series is completely reducible. This family contains, by a result of Reutenauer, the submonoids generated by bifix codes and, by a result of Berstel and Reutenauer, the cyclic sets. We study the closure properties of this family. We prove a result on linear representations of monoids which gives a generalization of the result concerning the complete reducibility of the submonoid generated by a bifix code to sets called birecurrent. We also give a new proof of the result concerning cyclic sets.

Permutation Cellular Automata

Permutation cellular automata are cellular automata defined by using finite maximal prefix codes. The overall dynamics of onesided and twosided permutation cellular automata is studied. For some classes of permutation cellular automata including the class of those defined by using finite maximal bifix codes, the overall dynamics is completely or partially described in terms of the codes used to define them.

Bifix codes and Sturmian words

We prove new results concerning the relation between bifix codes, episturmian words and subgroups of free groups. We study bifix codes in factorial sets of words. We generalize most properties of ordinary maximal bifix codes to bifix codes maximal in a recurrent set F of words (F-maximal bifix codes). In the case of bifix codes contained in Sturmian sets of words, we obtain several new results. Let F be a Sturmian set of words, defined as the set of factors of a strict episturmian word. Our results express the fact that an F-maximal bifix code of degree d behaves just as the set of words of F of length d. An F-maximal bifix code of degree d in a Sturmian set of words on an alphabet with k letters has (k−1)d+1 elements. This generalizes the fact that a Sturmian set contains (k−1)d+1 words of length d. Moreover, given an infinite word x, if there is a finite maximal bifix code X of degree d such that x has at most d factors of length d in X, then x is ultimately periodic. Our main result states that any F-maximal bifix code of degree d on the alphabet A is the basis of a subgroup of index d of the free group on A.

Some properties of involution codes

In this paper a generalization of the classical notions of codes, involution codes, are studied. This notion is motivated by DNA strand design where Watson-Crick complementarity can be modular as an antimorphic involution function. The characteristics of involution codes such as involution prefix codes and involution bifix codes are considered. The free property of the subfamilies of involution codes is studied as well. The author defines ρθ -languages and provides a procedure to construct ρθ -codes.

K-Comma Codes and Their Generalizations

In this paper, we introduce the notion of k-comma codes - a proper generalization of the notion of comma-free codes. For a given positive integer k, a k-comma code is a set L over an alphabet Σ with the property that LΣ kL ∩ Σ +LΣ + = ∅. Informally, in a k-comma code, no codeword can be a subword of the catenation of two other codewords separated by a “comma” of length k. A k-comma code is indeed a code, that is, any sequence of codewords is uniquely decipherable. We extend this notion to that of k-spacer codes, with commas of length less than or equal to a given k. We obtain several basic properties of k-comma codes and their generalizations, k-comma intercodes, and some relationships between the families of k-comma intercodes and other classical families of codes, such as infix codes and bifix codes. Moreover, we introduce the notion of n-k-comma intercodes, and obtain, for each k ≥ 0, several hierarchical relationships among the families of n-k-comma intercodes, as well as a characterization of the family of 1-k-comma intercodes.

Classifications of bifix codes

The family of all bifix codes is an important family of codes. A bifix code is a prefix code which is also a suffix code. In this paper we split the family of all bifix codes into three subfamilies, which are the family of all comma-codes, the family of all comma-free codes and the family of all strict intercodes of index m, m≥2. We study some combinatorial properties of them.

Annihilators of bifix codes

A bifix code is a prefix code which is also a suffix code. The family of all bifix codes is the most important family of all codes. This family has been split into subfamilies such as the family of comma codes, the family of comma-free codes and the families of strict intercodes of index m, m ≥ 2. This paper is a study of the properties of the annihilators of a given bifix code. The types of annihilators we will consider are comma code, comma-free code and strict intercode of index m annihilators. We present the existence of each kind of annihilators we considered. We obtained several interesting properties together with two-word comma codes and strict intercodes of index 2.

Completion of codes with finite bi-decoding delays

Let A∗ be a free monoid generated by a set A and let X⊆A∗ be a code with property P. The embedding of X into a complete code Y⊆A∗ with the same property P is called the completion of X. The method of completion of rational bifix codes and codes with finite decoding delays have been investigated by a number of authors. In this paper, we provide a general method of construction for completing the codes with finite bi-decoding delays. As a consequence, the completion method of rational bifix codes and codes with finite decoding delays is extended and applied to codes with finite bi-decoding delays.

Finite maximal solid codes

Solid 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.