Searching for Voltage Graph-Based LDPC Tailbiting Codes with Large Girth

The relation between parity-check matrices of quasi-cyclic (QC) low-density parity-check (LDPC) codes and biadjacency matrices of bipartite graphs supports searching for powerful LDPC block codes. Using the principle of tailbiting, compact representations of bipartite graphs based on convolutional codes can be found. Bounds on the girth and the minimum distance of LDPC block codes constructed in such a way are discussed. Algorithms for searching iteratively for LDPC block codes with large girth and for determining their minimum distance are presented. Constructions based on all-ones matrices, Steiner Triple Systems, and QC block codes are introduced. Finally, new QC regular LDPC block codes with girth up to 24 are given.Comment: Submitted to IEEE Transactions on Information Theory in February 201

Codes on Graphs and More

Modern communication systems strive to achieve reliable and efficient information transmission and storage with affordable complexity. Hence, efficient low-complexity channel codes providing low probabilities for erroneous receptions are needed. Interpreting codes as graphs and graphs as codes opens new perspectives for constructing such channel codes. Low-density parity-check (LDPC) codes are one of the most recent examples of codes defined on graphs, providing a better bit error probability than other block codes, given the same decoding complexity. After an introduction to coding theory, different graphical representations for channel codes are reviewed. Based on ideas from graph theory, new algorithms are introduced to iteratively search for LDPC block codes with large girth and to determine their minimum distance. In particular, new LDPC block codes of different rates and with girth up to 24 are presented. Woven convolutional codes are introduced as a generalization of graph-based codes and an asymptotic bound on their free distance, namely, the Costello lower bound, is proven. Moreover, promising examples of woven convolutional codes are given, including a rate 5/20 code with overall constraint length 67 and free distance 120. The remaining part of this dissertation focuses on basic properties of convolutional codes. First, a recurrent equation to determine a closed form expression of the exact decoding bit error probability for convolutional codes is presented. The obtained closed form expression is evaluated for various realizations of encoders, including rate 1/2 and 2/3 encoders, of as many as 16 states. Moreover, MacWilliams-type identities are revisited and a recursion for sequences of spectra of truncated as well as tailbitten convolutional codes and their duals is derived. Finally, the dissertation is concluded with exhaustive searches for convolutional codes of various rates with either optimum free distance or optimum distance profile, extending previously published results

Double-Hamming based QC LDPC codes with large minimum distance

A new method using Hamming codes to construct base matrices of (J, K)-regular LDPC convolutional codes with large free distance is presented. By proper labeling the corresponding base matrices and tailbiting these parent convolutional codes to given lengths, a large set of quasi-cyclic (QC) (J, K)-regular LDPC block codes with large minimum distance is obtained. The corresponding Tanner graphs have girth up to 14. This new construction is compared with two previously known constructions of QC (J, K)-regular LDPC block codes with large minimum distance exceeding (J+1)!. Applying all three constructions, new QC (J, K)-regular block LDPC codes with J=3 or 4, shorter codeword lengths and/or better distance properties than those of previously known codes are presented

New low-density parity-check codes with large girth based on hypergraphs

codes with large girth based on hypergraphs. In [Host publication title missing]. 10.1109/ISIT.2010.5513639 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 17. Sep. 2016 IEEE COPYRIGHT NOTICE c©2010 IEEE. Personal use of this material is permitted. Permission fro

A Greedy Search for Improved QC LDPC Codes with Good Girth Profile and Degree Distribution

Search algorithms for regular and irregular quasi-cyclic LDPC block codes with both good girth profile and good degree distribution are presented. New QC LDPC block codes of various code rates are obtained and their bit error rate performance is compared with that of the corresponding LDPC block codes defined in the IEEE 802.16 WiMAX standard of the same block length and code rate

Efficient Search of Compact QC-LDPC and SC-LDPC Convolutional Codes with Large Girth

We propose a low-complexity method to find quasi-cyclic low-density parity-check block codes with girth 10 or 12 and shorter length than those designed through classical approaches. The method is extended to time-invariant spatially coupled low-density parity-check convolutional codes, permitting to achieve small syndrome former constraint lengths. Several numerical examples are given to show its effectiveness.Comment: 4 pages, 3 figures, 1 table, accepted for publication in IEEE Communications Letter

Time-Invariant Spatially Coupled Low-Density Parity-Check Codes with Small Constraint Length

We consider a special family of SC-LDPC codes, that is, time-invariant LDPCC codes, which are known in the literature for a long time. Codes of this kind are usually designed by starting from QC block codes, and applying suitable unwrapping procedures. We show that, by directly designing the LDPCC code syndrome former matrix without the constraints of the underlying QC block code, it is possible to achieve smaller constraint lengths with respect to the best solutions available in the literature. We also find theoretical lower bounds on the syndrome former constraint length for codes with a specified minimum length of the local cycles in their Tanner graphs. For this purpose, we exploit a new approach based on a numerical representation of the syndrome former matrix, which generalizes over a technique we already used to study a special subclass of the codes here considered.Comment: 5 pages, 4 figures, to be presented at IEEE BlackSeaCom 201

Compact QC-LDPC Block and SC-LDPC Convolutional Codes for Low-Latency Communications

Low decoding latency and complexity are two important requirements of channel codes used in many applications, like machine-to-machine communications. In this paper, we show how these requirements can be fulfilled by using some special quasi-cyclic low-density parity-check block codes and spatially coupled low-density parity-check convolutional codes that we denote as compact. They are defined by parity-check matrices designed according to a recent approach based on sequentially multiplied columns. This method allows obtaining codes with girth up to 12. Many numerical examples of practical codes are provided.Comment: 5 pages, 1 figure, presented at IEEE PIMRC 201