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IEEE Information Theory Workshop (ITW)

Construction of LDPC codes for AWGN and binary erasure channels based on finite fields

Lingqi Zeng, Longwen Lan, Y.Y. Tai, Shu Lin, Khaled Abdel-Ghaffar

Published Jan 1, 2005• 4 citations
Publisher

Abstract

This paper presents a method for constructing structured LDPC codes based on finite fields and affine mappings. The Tanner graphs of these codes have girth at least six. Experimental results show that codes constructed based on this method with iterative decoding perform well on both the AWGN and binary erasure channels. Furthermore, codes constructed based on prime fields are quasi-cyclic.

DOI
10.1109/itw.2005.1531903

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This paper presents a method for constructing structured LDPC codes based on finite fields and affine mappings. The Tanner graphs of these codes have girth at least six. Experimental results show that codes constructed based on this method with iterative decoding perform well on both the AWGN and binary erasure channels. Furthermore, codes constructed based on prime fields are quasi-cyclic.
In the late 1950s and early 1960s, finite fields were successfully used to construct linear block codes, especially cyclic codes, with large minimum distances for hard-decision algebraic decoding, such as Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon (RS) codes. This paper shows that finite fields can also be successfully used to construct algebraic low-density parity-check (LDPC) codes for iterative soft-decision decoding. Methods of construction are presented. LDPC codes constructed by these methods are quasi-cyclic (QC) and they perform very well over the additive white Gaussian noise (AWGN), binary random, and burst erasure channels with iterative decoding in terms of bit-error probability, block-error probability, error-floor, and rate of decoding convergence, collectively. Particularly, they have low error floors. Since the codes are QC, they can be encoded using simple shift registers with linear complexity.
This paper presents an algebraic method for constructing structural nonbinary LDPC codes based on the prime difference over finite field. The approach gives a class of quasi-cyclic (QC)-LDPC codes with girth at least 6. Especially, the approach can construct a class of short length code and the decoding complexity is low. Simulation results show that the constructed codes decoded with iterative decoding perform well over the AWGN channel using BPSK transmission in terms of bit-error probability and rate of decoding convergence, collectively.
Quasi-cyclic codes are the most promising class of structured LDPC codes due to their ease of implementation and excellent performance over noisy channels when decoded with message-passing algorithms as extensive simulation studies have shown. An approach for constructing quasi-cyclic LDPC codes based on Latin squares over finite fields is presented. By analyzing the parity-check matrices of these codes, expressions for their ranks are derived. Experimental results show that, with iterative decoding algorithms, the constructed codes perform very well over the AWGN and the binary erasure channels.