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1 Design of regular (2, dc)-LDPC codes over GF(q) using their binary images

Charly Poulliat, M.P.C. Fossorier, David Declercq

Published Aug 24, 2013• 0 citations
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Abstract

Abstract — In this paper, a method to design regular (2, dc)-LDPC codes over GF(q) with both good waterfall and error floor properties is presented, based on the algebraic properties of their binary image. First, the algebraic properties of rows of the parity check matrix H associated with a code are characterized and optimized to improve the waterfall. Then the algebraic properties of cycles and stopping sets associated with the underlying Tanner graph are studied and linked to the global binary minimum distance of the code. Finally, simulations are presented to illustrate the excellent performance of the designed codes. Index Terms — channel coding, error correction coding, nonbinary LDPC codes, iterative decoding, binary image. I.

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Abstract — In this paper, a method to design regular (2, dc)-LDPC codes over GF(q) with both good waterfall and error floor properties is presented, based on the algebraic properties of their binary image. First, the algebraic properties of rows of the parity check matrix H associated with a code are characterized and optimized to improve the waterfall. Then the algebraic properties of cycles and stopping sets associated with the underlying Tanner graph are studied and linked to the global binary minimum distance of the code. Finally, simulations are presented to illustrate the excellent performance of the designed codes. Index Terms — channel coding, error correction coding, nonbinary LDPC codes, iterative decoding, binary image. I.
In this paper, a method to design regular (2, dc)- LDPC codes over GF(q) with both good waterfall and error floor properties is presented, based on the algebraic properties of their binary image. First, the algebraic properties of rows of the parity check matrix H associated with a code are characterized and optimized to improve the waterfall. Then the algebraic properties of cycles and stopping sets associated with the underlying Tanner graph are studied and linked to the global binary minimum distance of the code. Finally, simulations are presented to illustrate the excellent performance of the designed codes.
In order to construct nonbinary low-density parity-check(LDPC) codes with good waterfall and error floor properties,a method to improve the minimum Hamming distance of a class of nonbinary quasi-cyclic(QC) LDPC codes is presented.Two upper bounds on the minimum Hamming distance are derived for the nonbinary QC LDPC codes whose parity-check matrices consist of weight-2 columns.A special base matrix design allowing linear encoding in a parallel manner is also proposed.In addition,the design principals are given,according to which,the parallel and efficiently encodable codes reaching the upper bound on the minimum Hamming distance can be developed.Simulation shows that the proposed codes have both good waterfall and error floor properties.
Quasi-cyclic (QC) low-density parity-check (LDPC) codes have several appealing properties regarding decoding, storage requirements and encoding aspects. In this paper, we focus on the QC LDPC codes over GF(q) whose parity-check matrices have fixed column weight j=2. By investigating two subgraphs in the Tanner graphs of the corresponding base matrices, we derive two upper bounds on the minimum Hamming distance for this class of codes. In addition, a method is proposed to construct QC LDPC codes over GF(q), which have good Hamming distance distributions. Simulations show that our designed codes have good performance.
A (v,g)-cage is a (not necessarily unique) smallest v-regular graph of girth g. On such a graph, a nonbinary (2,v)-regular low-density parity-check (LDPC) code can be defined such that the Tanner graph has girth 2g and the code length achieves the minimum possible. In this paper, we focus on two aspects of this class of codes, structural property and code optimization. We find that, in addition to those found previously, many cages can be used to construct structured LDPC codes. We show that all cages with even girth can be structured as protograph-based codes, many of which have block-circulant Tanner graphs. We also find that four cages with odd girth can be structured as protograph-based codes with block-circulant Tanner graphs. For code optimization, we develop an ontology-based approach. All possible inter-connected cycle patterns that lead to low symbol-weight codewords are identified to put together the ontology. By doing so, it becomes handleable to estimate and optimize distance spectrum of equivalent binary image codes. We further analyze some known codes from the Consultative Committee for Space Data Systems recommendation and design several new codes. Numerical results show that these codes have reasonably good minimum bit distance and perform well under iterative decoding.