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Annals of Computer Science and Information Systems

On LDPC codes corresponding to infinite family of graphs A(k, K)

Monika Polak, Vasyl Ustimenko

Published Nov 20, 2012• 5 citations
OpenAlex

Abstract

In this paper we investigate correcting properties of LDPC error correcting codes obtained from new infinite family of special extremal graphs. We describe how to construct these codes and compare our results with codes obtained by Guinand and Lodge, corresponding to family of graphs D(k, q).

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In this paper we investigate correcting properties of LDPC codes obtained from families of algebraic graphs. The graphs considered in this article come from the infinite incidence structure. We describe how to construct these codes, choose the parameters and present several simulations, done by using the MAP decoder. We describe how error correcting properties are dependent on the graph structure. We compare our results with the currently used codes, obtained by Guinand and Lodge [1] from the family of graphs D(k; q), which were constructed by Ustimenko and Lazebnik [2].

Annales Universitatis Mariae Curie-Sklodowska sectio AI – Informatica•Jan 1, 2012•1 citation

GRAPH BASED LINEAR ERROR CORRECTING CODES

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In this article we present a construction of error correcting codes, that have representation as very sparse matrices and belong to the class of Low Density Parity Check Codes. LDPC codes are in the classical Hamming metric. They are very close to well known Shannon bound. The ability to use graphs for code construction was first discussed by Tanner in 1981 and has been used in a number of very effective implementations. We describe how to construct such codes by using special a family of graphs introduced by Ustimenko and Woldar. Graphs that we used are bipartite, bi-regular, very sparse and do not have short cycles C4. Due to the very low density of such graphs, the obtained codes are fast decodable. We describe how to choose parameters to obtain a desired code rate. We also show results of computer simulations of BER (bit error rate) of the obtained codes in order to compare them with other known LDPC codes.

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LDPC Codes from Deza Graphs

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We give a construction of LDPC codes from Deza graphs with parameters $(v,k,1,0)$ . The Tanner graphs of these LDPC codes do not contain cycles of length 4. Special attention is given to the construction of LDPC codes from the Moore graphs with diameter 2. We identify and enumerate the smallest absorbing sets in the Tanner graphs of the obtained LDPC codes and analyze their structures. Furthermore, we describe a construction, based on the protograph operation, of an infinite family of LDPC codes, whose Tanner graphs have girth at least 6, obtained from each $(v,k,1,0)$ Deza graph. We give an expression for the variance of a syndrome weight of the constructed LDPC codes, and also present simulation results.

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Low-density parity-check codes from combinatorial designs

Sarah J. Johnson

Abstract Recent breakthroughs in forward error correction, in the form of low-density parity-check(LDPC) and turbo codes, have seen the promise of near Shannon limit information trans-mission realised. The deterministic construction of good codes, for LDPC codes in par-ticular, is currently the subject of intense interest in the research and development com-munity. This thesis adds to this field, developing methods and supporting theory for thedesign of algebraic LDPC codes using incidence structures from combinatorial designs.The benefits of this approach are guaranteed code properties, parameter flexibility, easeof implementation and reduced encoding complexity.In the first part of this thesis Steiner 2-designs and partial geometries are used toprovide systematic constructions of regular LDPC codes with Tanner graphs free of 4-cyclesandwithdeterministiccodeproperties. Drawingextensivelyfromexistingincidencestructures we present new families of algebraic LDPC codes with benefits over traditionalrandomly constructed regular codes, particularly where small-to-medium length codes areconcerned. We focus in particular on exploring the effect of different design propertieson the code parameters by deriving the rate, minimum distance, girth, and stopping setdistribution of LDPC codes from designs.In the second part of this thesis resolvable designs are employed to create regularLDPC codes free of 4-cycles with the parameter flexibility and decoding performance ofrandomly constructed regular LDPC codes. Employing resolvability greatly extends theclass of LDPC codes that can be algebraically constructed. These codes can offer sig-nificant implementation advantages due to the deterministic structure of their resolutionclasses and in some cases their simple encoding.A shortcoming of LDPC codes is their potentially high encoding complexity, whichis generally quadratic in the code length. Finding computationally efficient encodersis therefore critical for LDPC codes to be considered as serious contenders for futuregenerations of forward error correction devices. As a result the purpose of the last part ofthis thesis is to design codes which have the graph-based properties necessary to performwell with sum-product decoding but which are also cyclic or quasi-cyclic and so can beencoded simply. The main focus is regular codes but we also extend our constructionof quasi-cyclic codes to produce irregular LDPC codes with simple linear-time encodingcircuits.

•Jan 1, 2004•14 citations

Analysis of the relation between properties of LDPC codes and the tanner graph

Victor Zyablov, Pavel S. Rybin

A new method for estimating the number of errors guaranteed to be corrected by a low-density parity-check code is proposed. The method is obtained by analyzing edges with special properties of an appropriate Tanner graph. In this paper we consider binary LDPC codes with constituent single-parity-check and Hamming codes and an iterative decoding algorithm. Numerical results obtained for the proposed lower bound exceed similar results for the best previously known lower bounds.

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Theory and Practice of Non-Binary Graph-Based Codes: A Combinatorial View

Behzad Amiri

We are undergoing a revolution in data. The ever-growing amount of information in our world has created an unprecedented demand for ultra-reliable, affordable, and resource-efficient data storage systems. Error-correcting codes, as a critical component of any memory device, will play a crucial role in the future of data storage.One particular class of error-correcting codes, known as graph-based codes, has drawn significant attention in both academia and in industry. Graph-based codes offer superior performance compared to traditional algebraic codes. Recently, it has been shown that non-binary graph-based codes, which operate over finite fields rather than binary alphabets, outperform their binary counterparts and exhibit outstanding overall performance. For this reason, these codes are particularly suitable for emerging data storage systems.In this dissertation, we present a comprehensive combinatorial analysis of non-binary graph-based codes. We perform both finite-length and asymptotic analyses for these codes, providing a systematic framework to evaluate and optimize various families of non-binary graph-based codes.In the finite-length case, we provide a mathematical characterization of the error floor problem, including a general definition of absorbing sets over non-binary alphabets. We consider several structured low-density parity-check (LDPC) codes, including quasi-cyclic and spatially-coupled codes, as well as unstructured LDPC codes. We offer design guidelines for non-binary LDPC codes with outstanding performance in extremely low error-rate regimes; making them excellent candidates for data storage applications.In the asymptotic case, we provide a novel toolbox for the evaluation of families of non-binary graph-based codes. By utilizing insights from graph theory and combinatorics, we establish enumerators for a general family of graph-based codes which are constructed based on protographs. We provide asymptotic distributions of codewords and trapping sets for the family of protograph-based codes. Furthermore, we present an asymptotic enumeration of binary and non-binary elementary absorbing sets for regular code ensembles.The contributions of this dissertation can potentially impact a broad range of data storage and communication technologies that require excellent performance in high-reliability regimes.

•Jan 1, 2015•0 citations
In this paper we investigate correcting properties of LDPC codes obtained from families of algebraic graphs. The graphs considered in this article come from the infinite incidence structure. We describe how to construct these codes, choose the parameters and present several simulations, done by using the MAP decoder. We describe how error correcting properties are dependent on the graph structure. We compare our results with the currently used codes, obtained by Guinand and Lodge [1] from the family of graphs D(k; q), which were constructed by Ustimenko and Lazebnik [2].
In this article we present a construction of error correcting codes, that have representation as very sparse matrices and belong to the class of Low Density Parity Check Codes. LDPC codes are in the classical Hamming metric. They are very close to well known Shannon bound. The ability to use graphs for code construction was first discussed by Tanner in 1981 and has been used in a number of very effective implementations. We describe how to construct such codes by using special a family of graphs introduced by Ustimenko and Woldar. Graphs that we used are bipartite, bi-regular, very sparse and do not have short cycles C4. Due to the very low density of such graphs, the obtained codes are fast decodable. We describe how to choose parameters to obtain a desired code rate. We also show results of computer simulations of BER (bit error rate) of the obtained codes in order to compare them with other known LDPC codes.
We give a construction of LDPC codes from Deza graphs with parameters (v,k,1,0)(v,k,1,0)(v,k,1,0) . The Tanner graphs of these LDPC codes do not contain cycles of length 4. Special attention is given to the construction of LDPC codes from the Moore graphs with diameter 2. We identify and enumerate the smallest absorbing sets in the Tanner graphs of the obtained LDPC codes and analyze their structures. Furthermore, we describe a construction, based on the protograph operation, of an infinite family of LDPC codes, whose Tanner graphs have girth at least 6, obtained from each (v,k,1,0)(v,k,1,0)(v,k,1,0) Deza graph. We give an expression for the variance of a syndrome weight of the constructed LDPC codes, and also present simulation results.
Abstract Recent breakthroughs in forward error correction, in the form of low-density parity-check(LDPC) and turbo codes, have seen the promise of near Shannon limit information trans-mission realised. The deterministic construction of good codes, for LDPC codes in par-ticular, is currently the subject of intense interest in the research and development com-munity. This thesis adds to this field, developing methods and supporting theory for thedesign of algebraic LDPC codes using incidence structures from combinatorial designs.The benefits of this approach are guaranteed code properties, parameter flexibility, easeof implementation and reduced encoding complexity.In the first part of this thesis Steiner 2-designs and partial geometries are used toprovide systematic constructions of regular LDPC codes with Tanner graphs free of 4-cyclesandwithdeterministiccodeproperties. Drawingextensivelyfromexistingincidencestructures we present new families of algebraic LDPC codes with benefits over traditionalrandomly constructed regular codes, particularly where small-to-medium length codes areconcerned. We focus in particular on exploring the effect of different design propertieson the code parameters by deriving the rate, minimum distance, girth, and stopping setdistribution of LDPC codes from designs.In the second part of this thesis resolvable designs are employed to create regularLDPC codes free of 4-cycles with the parameter flexibility and decoding performance ofrandomly constructed regular LDPC codes. Employing resolvability greatly extends theclass of LDPC codes that can be algebraically constructed. These codes can offer sig-nificant implementation advantages due to the deterministic structure of their resolutionclasses and in some cases their simple encoding.A shortcoming of LDPC codes is their potentially high encoding complexity, whichis generally quadratic in the code length. Finding computationally efficient encodersis therefore critical for LDPC codes to be considered as serious contenders for futuregenerations of forward error correction devices. As a result the purpose of the last part ofthis thesis is to design codes which have the graph-based properties necessary to performwell with sum-product decoding but which are also cyclic or quasi-cyclic and so can beencoded simply. The main focus is regular codes but we also extend our constructionof quasi-cyclic codes to produce irregular LDPC codes with simple linear-time encodingcircuits.
We are undergoing a revolution in data. The ever-growing amount of information in our world has created an unprecedented demand for ultra-reliable, affordable, and resource-efficient data storage systems. Error-correcting codes, as a critical component of any memory device, will play a crucial role in the future of data storage.One particular class of error-correcting codes, known as graph-based codes, has drawn significant attention in both academia and in industry. Graph-based codes offer superior performance compared to traditional algebraic codes. Recently, it has been shown that non-binary graph-based codes, which operate over finite fields rather than binary alphabets, outperform their binary counterparts and exhibit outstanding overall performance. For this reason, these codes are particularly suitable for emerging data storage systems.In this dissertation, we present a comprehensive combinatorial analysis of non-binary graph-based codes. We perform both finite-length and asymptotic analyses for these codes, providing a systematic framework to evaluate and optimize various families of non-binary graph-based codes.In the finite-length case, we provide a mathematical characterization of the error floor problem, including a general definition of absorbing sets over non-binary alphabets. We consider several structured low-density parity-check (LDPC) codes, including quasi-cyclic and spatially-coupled codes, as well as unstructured LDPC codes. We offer design guidelines for non-binary LDPC codes with outstanding performance in extremely low error-rate regimes; making them excellent candidates for data storage applications.In the asymptotic case, we provide a novel toolbox for the evaluation of families of non-binary graph-based codes. By utilizing insights from graph theory and combinatorics, we establish enumerators for a general family of graph-based codes which are constructed based on protographs. We provide asymptotic distributions of codewords and trapping sets for the family of protograph-based codes. Furthermore, we present an asymptotic enumeration of binary and non-binary elementary absorbing sets for regular code ensembles.The contributions of this dissertation can potentially impact a broad range of data storage and communication technologies that require excellent performance in high-reliability regimes.