A Method to determine Partial Weight Enumerator for Linear Block Codes

In this paper we present a fast and efficient method to find partial weight enumerator (PWE) for binary linear block codes by using the error impulse technique and Monte Carlo method. This PWE can be used to compute an upper bound of the error probability for the soft decision maximum likelihood decoder (MLD). As application of this method we give partial weight enumerators and analytical performances of the BCH(130,66), BCH(103,47) and BCH(111,55) shortened codes; the first code is obtained by shortening the binary primitive BCH (255,191,17) code and the two other codes are obtained by shortening the binary primitive BCH(127,71,19) code. The weight distributions of these three codes are unknown at our knowledge.Comment: Computer Engineering and Intelligent Systems Vol 3, No.11, 201

Modern Coding Theory: The Statistical Mechanics and Computer Science Point of View

These are the notes for a set of lectures delivered by the two authors at the Les Houches Summer School on `Complex Systems' in July 2006. They provide an introduction to the basic concepts in modern (probabilistic) coding theory, highlighting connections with statistical mechanics. We also stress common concepts with other disciplines dealing with similar problems that can be generically referred to as `large graphical models'. While most of the lectures are devoted to the classical channel coding problem over simple memoryless channels, we present a discussion of more complex channel models. We conclude with an overview of the main open challenges in the field.Comment: Lectures at Les Houches Summer School on `Complex Systems', July 2006, 44 pages, 25 ps figure

Construction of quasi-cyclic self-dual codes

There is a one-to-one correspondence between $\ell$-quasi-cyclic codes over a finite field $\mathbb F_q$ and linear codes over a ring $R = \mathbb F_q[Y]/(Y^m-1)$. Using this correspondence, we prove that every $\ell$-quasi-cyclic self-dual code of length $m\ell$ over a finite field $\mathbb F_q$ can be obtained by the {\it building-up} construction, provided that char $(\mathbb F_q)=2$ or $q \equiv 1 \pmod 4$, $m$ is a prime $p$, and $q$ is a primitive element of $\mathbb F_p$. We determine possible weight enumerators of a binary $\ell$-quasi-cyclic self-dual code of length $p\ell$ (with $p$ a prime) in terms of divisibility by $p$. We improve the result of [3] by constructing new binary cubic (i.e., $\ell$-quasi-cyclic codes of length $3\ell$) optimal self-dual codes of lengths $30, 36, 42, 48$ (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When $m=5$, we obtain a new 8-quasi-cyclic self-dual $[40, 20, 12]$ code over $\mathbb F_3$ and a new 6-quasi-cyclic self-dual $[30, 15, 10]$ code over $\mathbb F_4$. When $m=7$, we find a new 4-quasi-cyclic self-dual $[28, 14, 9]$ code over $\mathbb F_4$ and a new 6-quasi-cyclic self-dual $[42,21,12]$ code over $\mathbb F_4$.Comment: 25 pages, 2 tables; Finite Fields and Their Applications, 201

Error Correcting Coding for a Non-symmetric Ternary Channel

Ternary channels can be used to model the behavior of some memory devices, where information is stored in three different levels. In this paper, error correcting coding for a ternary channel where some of the error transitions are not allowed, is considered. The resulting channel is non-symmetric, therefore classical linear codes are not optimal for this channel. We define the maximum-likelihood (ML) decoding rule for ternary codes over this channel and show that it is complex to compute, since it depends on the channel error probability. A simpler alternative decoding rule which depends only on code properties, called \da-decoding, is then proposed. It is shown that \da-decoding and ML decoding are equivalent, i.e., \da-decoding is optimal, under certain conditions. Assuming \da-decoding, we characterize the error correcting capabilities of ternary codes over the non-symmetric ternary channel. We also derive an upper bound and a constructive lower bound on the size of codes, given the code length and the minimum distance. The results arising from the constructive lower bound are then compared, for short sizes, to optimal codes (in terms of code size) found by a clique-based search. It is shown that the proposed construction method gives good codes, and that in some cases the codes are optimal.Comment: Submitted to IEEE Transactions on Information Theory. Part of this work was presented at the Information Theory and Applications Workshop 200

Self-Dual Codes

Self-dual codes are important because many of the best codes known are of this type and they have a rich mathematical theory. Topics covered in this survey include codes over F_2, F_3, F_4, F_q, Z_4, Z_m, shadow codes, weight enumerators, Gleason-Pierce theorem, invariant theory, Gleason theorems, bounds, mass formulae, enumeration, extremal codes, open problems. There is a comprehensive bibliography.Comment: 136 page

New Analytical Bounds on the Probability of Code-Word Error for Convolution Codes with Viterbi Decoding

New analytical bounds are developed for the probability of code-word error in a communication system with convolutional coding and soft-decision Viterbi decoding. The bounds are applicable to communications in channels with asynchronous interferers which are modeled as independent, partial-time white Gaussian interference sources. This model is often used in simulations to reflect the circumstances encountered in many packet radio communication networks. The new results include both purely analytical bounds and offline-simulation-aided bounds that permit implementation of accurate communication-link models with much lower online computational and storage requirements than are required with traditional Monte Carlo simulations of link performance. They significantly improve the trade-off that has previously existed between model fidelity and simulation complexity in Monte Carlo simulations of large-scale wireless communication networks with links employing convolutional coding

Some combinatorial invariants determined by Betti numbers of Stanley-Reisner ideals

The papers in this thesis are not available in Munin: Paper 1: Trygve Johnsen, Jan Roksvold, Hugues Verdure (2014): 'Betti numbers associated to the facet ideal of a matroid', available in Bulletin of the Brazilian Mathematical Society 45 no. 4, 727-744 Paper 2: Trygve Johnsen, Jan Roksvold, Hugues Verdure (2014): 'A generalization of weight polynomials to matroids', manuscript Paper 3: Jan Roksvold, Hugues Verdure (2015): 'Betti numbers of skeletons', manuscriptThe thesis contains new results on the connection between the algebraic properties of certain ideals of a polynomial ring and properties of error-correcting linear codes, matroids and simplicial complexes. We demonstrate that the graded Betti numbers of the facet ideal of a matroid are determined by the Betti numbers of the blocks of the matroid. The extended weight enumerator of coding theory is generalized to matroids. We show that this generalization is equivalent to the Tutte-polynomial, and that the coefficients of this polynomial is determined by Betti numbers of the Stanley-Reisner ideal of the matroid and its elongations. The Betti numbers of the Stanley-Reisner ring of a skeleton of a simplicial complex is demonstrated to be an integral linear combination of the Betti numbers associated to the original complex