New Combinatorial Construction Techniques for Low-Density Parity-Check Codes and Systematic Repeat-Accumulate Codes

This paper presents several new construction techniques for low-density parity-check (LDPC) and systematic repeat-accumulate (RA) codes. Based on specific classes of combinatorial designs, the improved code design focuses on high-rate structured codes with constant column weights 3 and higher. The proposed codes are efficiently encodable and exhibit good structural properties. Experimental results on decoding performance with the sum-product algorithm show that the novel codes offer substantial practical application potential, for instance, in high-speed applications in magnetic recording and optical communications channels.Comment: 10 pages; to appear in "IEEE Transactions on Communications

On the resolutions of cyclic Steiner triple systems with small parameters

The paper presents useful invariants of resolutions of cyclic $STS(v)$ with $v\le 39$, namely of all resolutions of cyclic $STS(15)$, $STS(21)$ and $STS(27)$, of the resolutions with nontrivial automorphisms of cyclic $STS(33)$ and of resolutions with automorphisms of order $13$ of cyclic $STS(39)$

Low-Density Parity-Check Codes From Transversal Designs With Improved Stopping Set Distributions

This paper examines the construction of low-density parity-check (LDPC) codes from transversal designs based on sets of mutually orthogonal Latin squares (MOLS). By transferring the concept of configurations in combinatorial designs to the level of Latin squares, we thoroughly investigate the occurrence and avoidance of stopping sets for the arising codes. Stopping sets are known to determine the decoding performance over the binary erasure channel and should be avoided for small sizes. Based on large sets of simple-structured MOLS, we derive powerful constraints for the choice of suitable subsets, leading to improved stopping set distributions for the corresponding codes. We focus on LDPC codes with column weight 4, but the results are also applicable for the construction of codes with higher column weights. Finally, we show that a subclass of the presented codes has quasi-cyclic structure which allows low-complexity encoding.Comment: 11 pages; to appear in "IEEE Transactions on Communications

Conception Avancée des codes LDPC binaires pour des applications pratiques

The design of binary LDPC codes with low error floors is still a significant problem not fully resolved in the literature. This thesis aims to design optimal/optimized binary LDPC codes. We have two main contributions to build the LDPC codes with low error floors. Our first contribution is an algorithm that enables the design of optimal QC-LDPC codes with maximum girth and mini-mum sizes. We show by simulations that our algorithm reaches the minimum bounds for regular QC-LDPC codes (3, d c ) with low d c . Our second contribution is an algorithm that allows the design optimized of regular LDPC codes by minimizing dominant trapping-sets/expansion-sets. This minimization is performed by a predictive detection of dominant trapping-sets/expansion-sets defined for a regular code C(d v , d c ) of girth g t . By simulations on different rate codes, we show that the codes designed by minimizing dominant trapping-sets/expansion-sets have better performance than the designed codes without taking account of trapping-sets/expansion-sets. The algorithms we proposed are based on the generalized RandPEG. These algorithms take into account non-cycles seen in the case of quasi-cyclic codes to ensure the predictions.La conception de codes LDPC binaires avec un faible plancher d’erreurs est encore un problème considérable non entièrement résolu dans la littérature. Cette thèse a pour objectif la conception optimale/optimisée de codes LDPC binaires. Nous avons deux contributions principales pour la construction de codes LDPC à faible plancher d’erreurs. Notre première contribution est un algorithme qui permet de concevoir des codes QC-LDPC optimaux à large girth avec les tailles minimales. Nous montrons par des simulations que notre algorithme atteint les bornes minimales fixées pour les codes QC-LDPC réguliers (3, d c ) avec d c faible. Notre deuxième contribution est un algorithme qui permet la conception optimisée des codes LDPC réguliers en minimisant les trapping-sets/expansion-sets dominants(es). Cette minimisation s’effectue par une détection prédictive des trapping-sets/expansion-sets dominants(es) définies pour un code régulier C(d v , d c ) de girth gt . Par simulations sur des codes de rendement différent, nous montrons que les codes conçus en minimisant les trapping-sets/expansion-sets dominants(es) ont de meilleures performances que les codes conçus sans la prise en compte des trapping-sets/expansion-sets. Les algorithmes que nous avons proposés se basent sur le RandPEG généralisé. Ces algorithmes prennent en compte les cycles non-vus dans le cas des codes quasi-cycliques pour garantir les prédictions

Structural Design and Analysis of Low-Density Parity-Check Codes and Systematic Repeat-Accumulate Codes

The discovery of two fundamental error-correcting code families, known as turbo codes and low-density parity-check (LDPC) codes, has led to a revolution in coding theory and to a paradigm shift from traditional algebraic codes towards modern graph-based codes that can be decoded by iterative message passing algorithms. From then on, it has become a focal point of research to develop powerful LDPC and turbo-like codes. Besides the classical domain of randomly constructed codes, an alternative and competitive line of research is concerned with highly structured LDPC and turbo-like codes based on combinatorial designs. Such codes are typically characterized by high code rates already at small to moderate code lengths and good code properties such as the avoidance of harmful 4-cycles in the code's factor graph. Furthermore, their structure can usually be exploited for an efficient implementation, in particular, they can be encoded with low complexity as opposed to random-like codes. Hence, these codes are suitable for high-speed applications such as magnetic recording or optical communication. This thesis greatly contributes to the field of structured LDPC codes and systematic repeat-accumulate (sRA) codes as a subclass of turbo-like codes by presenting new combinatorial construction techniques and algebraic methods for an improved code design. More specifically, novel and infinite families of high-rate structured LDPC codes and sRA codes are presented based on balanced incomplete block designs (BIBDs), which form a subclass of combinatorial designs. Besides of showing excellent error-correcting capabilites under iterative decoding, these codes can be implemented efficiently, since their inner structure enables low-complexity encoding and accelerated decoding algorithms. A further infinite series of structured LDPC codes is presented based on the notion of transversal designs, which form another subclass of combinatorial designs. By a proper configuration of these codes, they reveal an excellent decoding performance under iterative decoding, in particular, with very low error-floors. The approach for lowering these error-floors is threefold. First, a thorough analysis of the decoding failures is carried out, resulting in an extensive classification of so-called stopping sets and absorbing sets. These combinatorial entities are known to be the main cause of decoding failures in the error-floor region over the binary erasure channel (BEC) and additive white Gaussian noise (AWGN) channel, respectively. Second, the specific code structures are exploited in order to calculate conditions for the avoidance of the most harmful stopping and absorbing sets. Third, powerful design strategies are derived for the identification of those code instances with the best error-floor performances. The resulting codes can additionally be encoded with low complexity and thus are ideally suited for practical high-speed applications. Further investigations are carried out on the infinite family of structured LDPC codes based on finite geometries. It is known that these codes perform very well under iterative decoding and that their encoding can be achieved with low complexity. By combining the latest findings in the fields of finite geometries and combinatorial designs, we generate new theoretical insights about the decoding failures of such codes under iterative decoding. These examinations finally help to identify the geometric codes with the most beneficial error-correcting capabilities over the BEC