An efficient algorithm for constructing nearly optimal prefix codes

A new algorithm for constructing nearly optimal prefix codes in the case of unequal letter costs and unequal probabilities is presented. A bound on the maximal deviation from the optimum is derived and numerical examples are given. The algorithm has running time O(t·n) where t is the number of letters and n is the number of probabilities

Invariance: a Theoretical Approach for Coding Sets of Words Modulo Literal (Anti)Morphisms

Let $A$ be a finite or countable alphabet and let $\theta$ be literal (anti)morphism onto $A^*$ (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under $\theta$ ($\theta$-invariant for short).We establish an extension of the famous defect theorem. Moreover, we prove that for the so-called thin $\theta$-invariant codes, maximality and completeness are two equivalent notions. We prove that a similar property holds in the framework of some special families of $\theta$-invariant codes such as prefix (bifix) codes, codes with a finite deciphering delay, uniformly synchronized codes and circular codes. For a special class of involutive antimorphisms, we prove that any regular $\theta$-invariant code may be embedded into a complete one.Comment: To appear in Acts of WORDS 201

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 echange transformations is closed under maximal bifix decoding.Comment: arXiv admin note: substantial text overlap with arXiv:1305.0127, arXiv:1308.539

Embedding a θ\theta-invariant code into a complete one

Let A be a finite or countable alphabet and let $\theta$ be a literal (anti-)automorphism onto A * (by definition, such a correspondence is determinated by a permutation of the alphabet). This paper deals with sets which are invariant under $\theta$ ($\theta$-invariant for short) that is, languages L such that $\theta$ (L) is a subset of L.We establish an extension of the famous defect theorem. With regards to the so-called notion of completeness, we provide a series of examples of finite complete $\theta$-invariant codes. Moreover, we establish a formula which allows to embed any non-complete $\theta$-invariant code into a complete one. As a consequence, in the family of the so-called thin $\theta$--invariant codes, maximality and completeness are two equivalent notions.Comment: arXiv admin note: text overlap with arXiv:1705.0556

On the group of a rational maximal bifix code

We give necessary and sufficient conditions for the group of a rational maximal bifix code $Z$ to be isomorphic with the $F$-group of $Z\cap F$, when $F$ is recurrent and $Z\cap F$ is rational. The case where $F$ is uniformly recurrent, which is known to imply the finiteness of $Z\cap F$, receives special attention. The proofs are done by exploring the connections with the structure of the free profinite monoid over the alphabet of $F$

Lossless quantum data compression and variable-length coding

In order to compress quantum messages without loss of information it is necessary to allow the length of the encoded messages to vary. We develop a general framework for variable-length quantum messages in close analogy to the classical case and show that lossless compression is only possible if the message to be compressed is known to the sender. The lossless compression of an ensemble of messages is bounded from below by its von-Neumann entropy. We show that it is possible to reduce the number of qbits passing through a quantum channel even below the von-Neumann entropy by adding a classical side-channel. We give an explicit communication protocol that realizes lossless and instantaneous quantum data compression and apply it to a simple example. This protocol can be used for both online quantum communication and storage of quantum data.Comment: 16 pages, 5 figure