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arXiv:1304.7487 · IEEE Information Theory Workshop (ITW)

Design of non-binary quasi-cyclic LDPC codes by ACE optimization

Alex Bazarsky, Noam Presman, Simon N. Litsyn

Published Sep 1, 2013• 22 citations
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Abstract

An algorithm for constructing Tanner graphs of non-binary irregular quasi-cyclic LDPC codes is introduced. It employs a new method for selection of edge labels allowing control over the code's non-binary ACE spectrum and resulting in low error-floor. The efficiency of the algorithm is demonstrated by generating good codes of short to moderate length over small fields, outperforming codes generated by the known methods.

DOI
10.1109/itw.2013.6691250

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An algorithm for constructing Tanner graphs of non-binary irregular quasi-cyclic LDPC codes is introduced. It employs a new method for selection of edge labels allowing control over the code's non-binary ACE spectrum and resulting in low error-floor. The efficiency of the algorithm is demonstrated by generating good codes of short to moderate length over small fields, outperforming codes generated by the known methods.
The construction of good, finite-length, LDPC codes is currently an attractive research area. Reducing attention to the binary erasure channel (BEC), this problem translates into the problem of finding elements of selected (irregular) LDPC code ensemble with the size of the minimal stopping set being maximized. Faced with the lack of analytical solution to this problem, simple but powerful heuristic design algorithm, an ACE constrained design algorithm, was recently introduced. Building upon the ACE metric associated with every cycle in the code graph, we introduce the ACE spectrum of LDPC code as a useful measure of success in selection of the LDPC code from selected ensemble. Using ACE spectrum, we farther generalize ACE constrained design, making it more flexible and efficient.
We propose a novel approach to optimization of irregular nonbinary (NB) quasi-cyclic (QC)-LDPC codes over small alphabets. In this approach, first, the base parity-check matrices are constructed by a simulated annealing method, and then these matrices are labeled by the field elements while maximizing the so- called generalized girth of the Tanner graph. In order to analyze the performance of the constructed irregular NB LDPC codes, a new ensemble of irregular NB LDPC codes over the extensions of the binary Galois field is introduced. A finite-length random coding bound on the error probability of the maximum-likelihood (ML) decoding over the binary phase shift keying (BPSK) input AWGN channel for the new code ensemble is derived. The frame error rate (FER) performance of the sum-product belief-propagation (BP) decoding of the constructed NB QC-LDPC block codes is compared to that of both the optimized binary QC-LDPC block codes in the 5G standard and the best known NB QC-LDPC codes as well as to the derived random coding bound on the ML decoding error probability. It is shown that the obtained bound predicts the behavior of BP decoding performance of practical NB QC-LDPC codes more accurately than the BP decoding thresholds do.
We propose a technique to design finite-length irregular low-density parity-check (LDPC) codes over the binary-input additive white Gaussian noise (AWGN) channel with good performance in both the waterfall and the error floor region. The design process starts from a protograph which embodies a desirable degree distribution. This protograph is then lifted cyclically to a certain block length of interest. The lift is designed carefully to satisfy a certain approximate cycle extrinsic message degree (ACE) spectrum. The target ACE spectrum is one with extremal properties, implying a good error floor performance for the designed code. The proposed construction results in quasi-cyclic codes which are attractive in practice due to simple encoder and decoder implementation. Simulation results are provided to demonstrate the effectiveness of the proposed construction in comparison with similar existing constructions.
Quasi-cyclic low-density parity-check (QC-LDPC) codes constructed by using algebraic approaches, such as finite geometry and finite field, generally have good structural properties and very low error-floors, and facilitate hardware implementation. Irregular QC-LDPC codes constructed from such QC-LDPC codes by using the masking technique, when decoded with the sum-product algorithm (SPA), have low decoding complexity, but often show early error-floors. In this paper, the relationship of cycle, girth and approximate cycle EMD (ACE) between the masking matrix and masked matrix is investigated, and the ACE algorithm is modified and used to construct the masking matrix for irregular QC-LDPC codes. Simulations demonstrate that the codes constructed by masking with ACE optimization exhibit much improved waterfall performance and lower error floors.