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arXiv:1007.0404 · IEEE Information Theory Workshop (ITW)

Quasi-cyclic asymptotically regular LDPC codes

David G. M. Mitchell, Roxana Smarandache, Michael Lentmaier, Daniel J. Costello

Published Aug 1, 2010• 8 citations
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Abstract

Families of asymptotically regular LDPC block code ensembles can be formed by terminating (J, K)-regular protograph-based LDPC convolutional codes. By varying the termination length, we obtain a large selection of LDPC block code ensembles with varying code rates, minimum distance that grows linearly with block length, and capacity approaching iterative decoding thresholds, despite the fact that the terminated ensembles are almost regular. In this paper, we investigate the properties of the quasi-cyclic (QC) members of such an ensemble. We show that an upper bound on the minimum Hamming distance of members of the QC sub-ensemble can be improved by careful choice of the component protographs used in the code construction. Further, we show that the upper bound on the minimum distance can be improved by using arrays of circulants in a graph cover of the protograph.

DOI
10.1109/cig.2010.5592641

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Families of asymptotically regular LDPC block code ensembles can be formed by terminating (J, K)-regular protograph-based LDPC convolutional codes. By varying the termination length, we obtain a large selection of LDPC block code ensembles with varying code rates, minimum distance that grows linearly with block length, and capacity approaching iterative decoding thresholds, despite the fact that the terminated ensembles are almost regular. In this paper, we investigate the properties of the quasi-cyclic (QC) members of such an ensemble. We show that an upper bound on the minimum Hamming distance of members of the QC sub-ensemble can be improved by careful choice of the component protographs used in the code construction. Further, we show that the upper bound on the minimum distance can be improved by using arrays of circulants in a graph cover of the protograph.
Families of asymptotically regular LDPC block code ensembles can be formed by terminating (J, K)-regular protograph-based LDPC convolutional codes. By varying the termination length, we obtain a large selection of LDPC block code ensembles with varying code rates and substantially better iterative decoding thresholds than those of (J, K)-regular LDPC block code ensembles, despite the fact that the terminated ensembles are almost regular. Also, by means of an asymptotic weight enumerator analysis, we show that minimum distance grows linearly with block length for all of the ensembles in these families, i.e., the ensembles are asymptotically good. We find that, as the termination length increases, families of ¿asymptotically regular¿ codes with capacity approaching iterative decoding thresholds and declining minimum distance growth rates are obtained, allowing a code designer to trade-off between distance growth rate and threshold. Further, we show that the thresholds and the distance growth rates can be improved by carefully choosing the component protographs used in the code construction.
A significant amount of the analysis of protograph-based low-density parity-check (LDPC) codes has been devoted to the subclass of quasi-cyclic (QC) LDPC codes. Despite their implementation advantages and algebraic properties that make them easy to analyze, protograph-based QC-LDPC codes have undesirable fixed upper limits on important code parameters. This implies that picking a QC code from an asymptotically good or capacity approaching ensemble is suboptimal, since long QC codes will not perform close to the ensemble asymptotic limits. Indeed, these limits can only be achieved by codes that are not QC. In this paper we present an overview together with some new results on partially-QC protograph-based LDPC codes, i.e., LDPC codes whose parity-check matrix is partially composed of circulant submatrices. We perform both a minimum Hamming distance and girth analysis of these codes. Moreover, we present explicit partially-QC LDPC code constructions with parameters that exceed the restricted QC upper bounds.
Recent work by Divsalar et al. has shown that properly designed protograph-based low-density parity-check (LDPC) codes typically have minimum (Hamming) distance linearly increasing with block length. This fact rests on ensemble arguments over all possible expansions of the base protograph. However, when implementation complexity is considered, the expansions are frequently selected from a smaller class of structured expansions. For example, protograph expansion by cyclically shifting connections generates a quasi-cyclic (QC) code. Other recent work by Smarandache and Vontobel has provided upper bounds on the minimum distance of QC codes. In this paper, we generalize these bounds to punctured QC codes and then show how to tighten these for certain classes of codes. We then evaluate these upper bounds for the family of protograph codes known as AR4JA codes that have been recommended for use in deep space communications in a standard established by the Consultative Committee for Space Data Systems (CCSDS). At block lengths larger than 4400 bits, these upper bounds fall well below the ensemble lower bounds.
In this paper, we present a method of constructing new families of LDPC block code ensembles formed by terminating irregular protograph-based LDPC convolutional codes. Using the accumulate-repeat-by-4-jagged-accumulate (AR4JA) protograph as an example, a density evolution analysis for the binary erasure channel shows that this flexible design technique gives rise to a large selection of LDPC block code ensembles with varying code rates and thresholds close to capacity. Further, by means of an asymptotic weight enumerator analysis, we show that all the ensembles in this family also have minimum distance that grows linearly with block length, i.e., they are asymptotically good.
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.
Recent work by Divsalarhas shown that properly designed protograph-based low-density parity-check codes typically have minimum (Hamming) distance linearly increasing with block length. This fact rests on ensemble arguments over all possible expansions of the base protograph. However, when implementation complexity is considered, the expansions are frequently selected from a smaller class of structured expansions. For example, protograph expansion by cyclically shifting connections generates a quasi-cyclic (QC) code. Other recent work by Smarandache and Vontobel has provided upper bounds on the minimum distance of QC codes. In this paper, we generalize these bounds to punctured QC codes and then show how to tighten these for certain classes of codes. We then evaluate these upper bounds for the family of protograph codes known as AR4JA codes that have been recommended for use in deep space communications in a standard established by the Consultative Committee for Space Data Systems. At block lengths larger than 4400 bits, these upper bounds fall well below the ensemble lower bounds.