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Information Theory and Applications Workshop (ITA)

Reed-Solomon based globally coupled quasi-cyclic LDPC codes

Juane Li, Keke Liu, Shu Lin, Khaled Abdel-Ghaffar

Published Feb 1, 2017• 20 citations
Publisher

Abstract

This paper presents a special type of LDPC codes constructed based on the conventional parity-check matrices of Reed-Solomon (RS) codes. LDPC codes of this type are referred to as globally coupled RS-LDPC codes. These codes are designed for correcting random errors, random short phased bursts of erasures and long bursts of erasures. The Tanner graph of a globally coupled RS-LDPC is composed of a number of disjoint copies of the Tanner graph of a local RS-LDPC code which are connected together by a group of overall check-nodes (CNs), called global CNs. A globally coupled RS-LDPC code can be decoded with a two-phase local/global iterative scheme which allows correction of local random errors and phased bursts of erasures in the local phase and global random errors and a single long burst of erasures in the global phase.

DOI
10.1109/ita.2017.8023442

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This paper presents a special type of LDPC codes constructed based on the conventional parity-check matrices of Reed-Solomon (RS) codes. LDPC codes of this type are referred to as globally coupled RS-LDPC codes. These codes are designed for correcting random errors, random short phased bursts of erasures and long bursts of erasures. The Tanner graph of a globally coupled RS-LDPC is composed of a number of disjoint copies of the Tanner graph of a local RS-LDPC code which are connected together by a group of overall check-nodes (CNs), called global CNs. A globally coupled RS-LDPC code can be decoded with a two-phase local/global iterative scheme which allows correction of local random errors and phased bursts of erasures in the local phase and global random errors and a single long burst of erasures in the global phase.
This paper presents a special type of LDPC codes with a structure related to but different from that of the spatially coupled LDPC codes. For an LDPC code of this type, its Tanner graph is composed of a set of small disjoint Tanner graphs which are connected together by a group of overall check-nodes, called global check-nodes. Codes of this type are called globally coupled LDPC codes and they perform well over both the additive Gaussian white noise and the binary-erasure channels. Furthermore, they are very effective at correcting erasures clustered in bursts. Two algebraic methods are presented for constructing these codes. A two-phase local/global iterative scheme for decoding these codes is presented. This decoding scheme allows correction of local random and global errors and/or erasures in two phases.
Designs and constructions of quasi-cyclic (QC) LDPC codes for the AWGN channel are presented. The codes are constructed based on the conventional parity-check matrices of Reed-Solomon (RS) codes and are referred to as RS-QC-LDPC codes. Several classes of RS-QC-LDPC codes are given. Cycle structural properties of the Tanner graphs of codes in these classes are analyzed and specific methods for constructing codes with girth at least eight and reducing their short cycles are presented. The designed codes perform well in both waterfall and low error-rate regions.
This paper presents designs and constructions of two classes of binary quasi-cyclic LDPC codes for correcting multiple random phased-bursts of erasures over the binary erasure channel. The erasure correction of codes in both classes is characterized by the cycle and adjacency structure of their Tanner graphs. Erasure correction of these codes is a very simple process which requires only modulo-2 additions. The codes in the second class are capable of correcting locally and globally distributed phased-bursts of erasures with a two-phase iterative erasure-correction process.
In this paper, we first propose a general framework for constructing quasi-cyclic low-density parity-check (QC-LDPC) codes based on a two-dimensional (2-D) maximum distance separable (MDS) code. Two classes of QC-LDPC codes are defined, whose parity-check matrices are transposes of each other. We then use a 2-D generalized Reed-Solomon (GRS) code to give a concrete construction. The decoding parity-check matrices have a large number of redundant parity-check equations while their Tanner graphs have a girth of at least 6. The minimum distances of the codes are very respectable as far as LDPC codes are concerned. We further show that many existing constructions of QC-LDPC codes based on finite fields in the literature can be unified under this construction. Experimental studies show that the constructed QC-LDPC codes perform well with the sum-product algorithm (SPA).