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International Conference on Communications

A Class of Low-Density Parity-Check Convolutional Codes Based on Difference Families

Yucheng He, Christian Cardinal, David Haccoun

Published Jan 1, 2008• 2 citations
Publisher

Abstract

An algebraic construction for a class of low-density parity-check (LDPC) convolutional codes is presented on the basis of difference families associated with the code generator matrix. It can be shown that these codes have girth of at least 10 on the Tanner graph which is independent of either the size of the code generator matrix or the minimum Hamming distance of the codes. The code construction guarantees the independence of the messages exchanged in the belief propagation decoding process during two successive decoding iterations. Computer simulations show that over the additive white Gaussian noise channel, the best error performance of these codes at moderate signal-to- noise ratio values is practically obtained using only three to five iterations.

DOI
10.1109/icc.2008.228

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An algebraic construction for a class of low-density parity-check (LDPC) convolutional codes is presented on the basis of difference families associated with the code generator matrix. It can be shown that these codes have girth of at least 10 on the Tanner graph which is independent of either the size of the code generator matrix or the minimum Hamming distance of the codes. The code construction guarantees the independence of the messages exchanged in the belief propagation decoding process during two successive decoding iterations. Computer simulations show that over the additive white Gaussian noise channel, the best error performance of these codes at moderate signal-to- noise ratio values is practically obtained using only three to five iterations.
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Low-density parity-check (LDPC) block codes invented by Gallager in 1962 achieve exceptional error performance on a wide variety of communication channels. LDPC convolutional codes are the convolutional counterparts of LDPC block codes. This dissertation describes techniques for the design of LDPC convolutional codes, and analyzes their distance properties and iterative decoding convergence thresholds. The construction of time invariant LDPC convolutional codes by unwrapping the Tanner graph of algebraically constructed quasi-cyclic LDPC codes is described. The convolutional codes outperform the quasi-cyclic codes from which they are derived. The design of parity-check matrices for time invariant LDPC convolutional codes by the polynomial extension of a base matrix is proposed. An upper bound on free distance, proving that time invariant LDPC codes are not asymptotically good, is obtained. The Tanner graph is used to describe a pipelined message passing based iterative decoder for LDPC convolutional codes that outputs decoding results continuously. The performance of LDPC block and convolutional codes are compared for fixed decoding parameters like computational complexity, processor complexity, and decoding delay. In each case, the LDPC convolutional code performance is at least as good as that of LDPC block codes. An analog circuit to implement pipelined decoding of LDPC convolutional codes is proposed. The distance properties of a permutation matrix based (time varying) ensemble of (J,K) regular LDPC convolutional codes is investigated. It is proved that these codes (for J > 2) have free distance increasing linearly with constraint length, i.e., they are asymptotically good. Further, the asymptotic free distance to constraint length ratio for the convolutional codes is several times larger than the minimum distance to block length ratio for corresponding LDPC block codes. Iterative decoding of terminated LDPC convolutional codes, based on the ensemble mentioned above, is analyzed, assuming transmission over a binary erasure channel. The structured irregularity of the codes leads to significantly better convergence thresholds compared to corresponding LDPC block codes. At the calculated thresholds, both the bit and block error probability can be made arbitrarily small. The results obtained suggest that the thresholds approach capacity with increasing J.
A class of algebraically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes and their convolutional counterparts is presented. The QC codes are described by sparse parity-check matrices comprised of blocks of circulant matrices. The sparse parity-check representation allows for practical graph-based iterative message-passing decoding. Based on the algebraic structure, bounds on the girth and minimum distance of the codes are found, and several possible encoding techniques are described. The performance of the QC LDPC block codes compares favorably with that of randomly constructed LDPC codes for short to moderate block lengths. The performance of the LDPC convolutional codes is superior to that of the QC codes on which they are based; this performance is the limiting performance obtained by increasing the circulant size of the base QC code. Finally, a continuous decoding procedure for the LDPC convolutional codes is described.
This correspondence presents a construction of quasicyclic (QC) low-density parity-check (LDPC) codes based on a special type of combinatorial designs known as block disjoint difference families (BDDFs). The proposed construction of QC-LDPC codes gives parity-check matrices with column weight three and Tanner graphs having a girth lower-bounded by 6. The proposed QC-LDPC codes provide an excellent performance with iterative decoding over an additive white Gaussian-noise (AWGN) channel. Performance analysis shows that the proposed short and moderate length QC-LDPC codes perform as well as their competitors in the lower signal-to-noise ratio (SNR) region but outperform in the higher SNR region. Also, the codes constructed are quasicyclic in nature, so the encoding can be done with simple shift-register circuits with linear complexity.
Due to their capacity approaching performance low-density parity-check (LDPC) codes gained a lot of attention in the last years. The parity-check matrix of the codes can be associated with a bipartite graph, called Tanner graph. To decrease the probability of decoding failure it is desirable to have LDPC codes with large girth of the associated Tanner graph. Moreover, to store such codes efficiently, it is desirable to have compact constructions for them. In this paper, we present constructions of LDPC convolutional codes with girth up to 121212 using a special class of Latin squares and several lifting steps, which enables a compact representation of these codes. With these techniques, we can provide constructions for well-performing and efficiently storable time-varying and time-invariant LDPC convolutional codes as well as for LDPC block codes.
An elementary combinatorial Tanner graph construction for a family of near-regular low density parity check (LDPC) codes achieving high girth is presented. These codes are near regular in the sense that the degree of a left/right vertex is allowed to differ by at most one from the average. The construction yields in quadratic time complexity an asymptotic code family with provable lower bounds on the rate and the girth for a given choice of block length and average degree. The construction gives flexibility in the choice of design parameters of the code like rate, girth and average degree. Performance simulations of iterative decoding algorithm for the AWGN channel on codes designed using the method demonstrate that these codes perform better than regular PEG codes and MacKay codes of similar length for all values of Signal to noise ratio.