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

Construction of Near-Optimum Burst Erasure Correcting Low-Density Parity-Check Codes

In this paper, a simple, general-purpose and effective tool for the design of low-density parity-check (LDPC) codes for iterative correction of bursts of erasures is presented. The design method consists in starting from the parity-check matrix of an LDPC code and developing an optimized parity-check matrix, with the same performance on the memory-less erasure channel, and suitable also for the iterative correction of single bursts of erasures. The parity-check matrix optimization is performed by an algorithm called pivot searching and swapping (PSS) algorithm, which executes permutations of carefully chosen columns of the parity-check matrix, after a local analysis of particular variable nodes called stopping set pivots. This algorithm can be in principle applied to any LDPC code. If the input parity-check matrix is designed for achieving good performance on the memory-less erasure channel, then the code obtained after the application of the PSS algorithm provides good joint correction of independent erasures and single erasure bursts. Numerical results are provided in order to show the effectiveness of the PSS algorithm when applied to different categories of LDPC codes.Comment: 15 pages, 4 figures. IEEE Trans. on Communications, accepted (submitted in Feb. 2007

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

A Simplified Min-Sum Decoding Algorithm for Non-Binary LDPC Codes

Non-binary low-density parity-check codes are robust to various channel impairments. However, based on the existing decoding algorithms, the decoder implementations are expensive because of their excessive computational complexity and memory usage. Based on the combinatorial optimization, we present an approximation method for the check node processing. The simulation results demonstrate that our scheme has small performance loss over the additive white Gaussian noise channel and independent Rayleigh fading channel. Furthermore, the proposed reduced-complexity realization provides significant savings on hardware, so it yields a good performance-complexity tradeoff and can be efficiently implemented.Comment: Partially presented in ICNC 2012, International Conference on Computing, Networking and Communications. Accepted by IEEE Transactions on Communication

Fast-Decodable Asymmetric Space-Time Codes from Division Algebras

Multiple-input double-output (MIDO) codes are important in the near-future wireless communications, where the portable end-user device is physically small and will typically contain at most two receive antennas. Especially tempting is the 4 x 2 channel due to its immediate applicability in the digital video broadcasting (DVB). Such channels optimally employ rate-two space-time (ST) codes consisting of (4 x 4) matrices. Unfortunately, such codes are in general very complex to decode, hence setting forth a call for constructions with reduced complexity. Recently, some reduced complexity constructions have been proposed, but they have mainly been based on different ad hoc methods and have resulted in isolated examples rather than in a more general class of codes. In this paper, it will be shown that a family of division algebra based MIDO codes will always result in at least 37.5% worst-case complexity reduction, while maintaining full diversity and, for the first time, the non-vanishing determinant (NVD) property. The reduction follows from the fact that, similarly to the Alamouti code, the codes will be subsets of matrix rings of the Hamiltonian quaternions, hence allowing simplified decoding. At the moment, such reductions are among the best known for rate-two MIDO codes. Several explicit constructions are presented and shown to have excellent performance through computer simulations.Comment: 26 pages, 1 figure, submitted to IEEE Trans. Inf. Theory, October 201