Shortened Array Codes of Large Girth

One approach to designing structured low-density parity-check (LDPC) codes with large girth is to shorten codes with small girth in such a manner that the deleted columns of the parity-check matrix contain all the variables involved in short cycles. This approach is especially effective if the parity-check matrix of a code is a matrix composed of blocks of circulant permutation matrices, as is the case for the class of codes known as array codes. We show how to shorten array codes by deleting certain columns of their parity-check matrices so as to increase their girth. The shortening approach is based on the observation that for array codes, and in fact for a slightly more general class of LDPC codes, the cycles in the corresponding Tanner graph are governed by certain homogeneous linear equations with integer coefficients. Consequently, we can selectively eliminate cycles from an array code by only retaining those columns from the parity-check matrix of the original code that are indexed by integer sequences that do not contain solutions to the equations governing those cycles. We provide Ramsey-theoretic estimates for the maximum number of columns that can be retained from the original parity-check matrix with the property that the sequence of their indices avoid solutions to various types of cycle-governing equations. This translates to estimates of the rate penalty incurred in shortening a code to eliminate cycles. Simulation results show that for the codes considered, shortening them to increase the girth can lead to significant gains in signal-to-noise ratio in the case of communication over an additive white Gaussian noise channel.Comment: 16 pages; 8 figures; to appear in IEEE Transactions on Information Theory, Aug 200

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

Use of the LDPC codes Over the Binary Erasure Multiple Access Channel

Wireless communications use different orthogonal multiple access techniques to access a radio spectrum. The need for the bandwidth efficiency and data rate enhancing increase with the tremendous growth in the number of mobile users. One promising solution to increase the data rate without increasing the bandwidth is non-orthogonal multiple access channel. For the noiseless channel like the data network, the non-orthogonal multiple access channel is named: Binary Erasure Multiple Access Channel (BEMAC). To achieve two corner points on the boundary region of the BEMAC, a half rate code is needed. One practical code which has good performance over the BEMAC is the Low Density Parity Check (LDPC) codes. The LDPC codes receive a lot of attention nowadays, due to the good performance and low decoding complexity. However, there is a tradeoff between the performance and the decoding complexity of the LDPC codes. In addition, the LDPC encoding complexity is a problem, because an LDPC code is defined with its parity check matrix which is sparse and random and lacks of structure. This thesis consists of two main parts. In the first part, we propose a new practical method to construct an irregular half LDPC code which has low encoding complexity. The constructed code supposed to have a good performance and low encoding complexity. To have a low encoding complexity, the parity check matrix of the code must have lower triangular shape. By implementing the encoder and the decoder, the performance of the code can be also evaluated. Due to the short cycles in the code and finite length of the code the actual rate of the code is degraded. To improve the actual rate of the code, the guessing algorithm is applied if the Belief Propagation is stuck. The actual rate of the code increases from 0.418 to0.44. The decoding complexity is not considered when the code is constructed. Next in the second part, a regular LDPC code is constructed which has low decoding complexity. The code is generated based on the Gallager method. We present a new method to improve the performance of an existing regular LDPC code. The proposed method does not add a high complexity to the decoder. The method uses a combination of three algorithms: 1- Standard Belief Propagation 2- Generalized tree-expected propagation 3- Guessing algorithm. The guessing algorithm is impractical when the number of guesses increases. Because the number of possibilities increases exponentially with increasing the number of guesses. A new guessing algorithm is proposed in this thesis. The new guessing algorithm reduces the number of possibilities by guessing on the variable nodes which are connected to a set of check nodes. The actual rate of the code increases from 0.41 to 0.43 after applying the proposed method and considering the number of possibilities equal to two in the new guessing algorithm

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum