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Canadian Workshop on Information Theory (CWIT)

Cycle and Distance Properties of Structured LDPC Codes Based on Circulant Permutation Matrices

Jean‐Baptiste Doré, Marie‐Hélène Hamon, Pierre Penard

Published Jun 1, 2007• 2 citations
Publisher

Abstract

This paper investigates cycle and distance properties of structured Low Density Parity Check (LDPC) codes based on circulant permutation matrices. A very simple algorithm to detect and construct codes with good cycle properties is proposed. A geometrical approach to design codes with good girth properties is then presented and illustrated, through the definition of rules for the design of parity check matrices without short length cycles. An upper bound on the minimal distance of the proposed structured code is also derived. The information provided by this bound can help the construction of the parity check matrices, in order to achieve a target minimum distance.

DOI
10.1109/cwit.2007.375686

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This paper investigates cycle and distance properties of structured Low Density Parity Check (LDPC) codes based on circulant permutation matrices. A very simple algorithm to detect and construct codes with good cycle properties is proposed. A geometrical approach to design codes with good girth properties is then presented and illustrated, through the definition of rules for the design of parity check matrices without short length cycles. An upper bound on the minimal distance of the proposed structured code is also derived. The information provided by this bound can help the construction of the parity check matrices, in order to achieve a target minimum distance.
In this correspondence, the construction of low-density parity-check (LDPC) codes from circulant permutation matrices is investigated. It is shown that such codes cannot have a Tanner graph representation with girth larger than 12, and a relatively mild necessary and sufficient condition for the code to have a girth of 6, 8,10, or 12 is derived. These results suggest that families of LDPC codes with such girth values are relatively easy to obtain and, consequently, additional parameters such as the minimum distance or the number of redundant check sums should be considered. To this end, a necessary condition for the codes investigated to reach their maximum possible minimum Hamming distance is proposed.
Cycle distribution of LDPC(Low-Density Parity-Check) codes affects the codes decoding performance and encoding complexity,however it is commonly NP hard to analyse.We propose the rotation-distance for analysis of QC-LDPC(Quasi-Cyclic Low-Density Parity-Check) code based on circulant matrices.The circulant sub-matrices within the parity-check are treated as a matrix node to simplify theTanner graphs of the codes.Thus cycles of QC-LDPC codes can be found efficiently,and we demonstrate the usefulness of the new method by a simple proof of the known result that 12 is an upper limit of the girth of the QC-LDPC codes we considered.Moreover,the cycle analysis based on the new method also reveals relations between decoding performance and the cycle distribution of the code.
We investigate the minimum distance of structured binary Low-Density Parity-Check (LDPC) codes whose parity-check matrices are of the form [C∣M][\mathbf{C} \vert \mathbf{M}][C∣M] where C\mathbf{C}C is circulant and of column weight 222, and M\mathbf{M}M has fixed column weight r≥3r \geq 3r≥3 and row weight at least 111. These codes are of interest because they are LDPC codes which come with a natural linear-time encoding algorithm. We show that the minimum distance of these codes is in O(nr−2r−1+ε)O(n^{\frac{r-2}{r-1} + ε})O(nr−1r−2​+ε), where nnn is the code length and ε>0ε> 0ε>0 is arbitrarily small. This improves the previously known upper bound in O(nr−1r)O(n^{\frac{r-1}{r}})O(nrr−1​) on the minimum distance of such codes.
The presence of circles would lay a great impact on the quasi-cyclic LDPC decoding.Therefore,the parity check matrix with good girth distribution is beneficial to the performance of code.This paper first makes a analysis on girth distribution of the check matrix.Then a reasonable constraint is set between the mother matrix and shift matrix,thus,by this way,the check matrix could maximize the minimum cycle length of the check matrix and the average length of the cycle.Finally,a method to construct QC-LDPC code with high girth is given.Simulation results show that this method for minimizing the short code cycle could construct good practical codes.
Short cycles in the Tanner graph of a given LDPC code has negative effect on the performance of iterative decoding. To improve the performance, we presented an approach for constructing Quasi-Cyclic Low-Density Parity-Check (QC-LDPC) codes whose associated Tanner graph has large girth and a small amount of short cycles. Each column block of the parity check matrix was designed by adding circulant permutation sub-matrices one after another, provided the current parity check matrix has minimal number of short cycles. The condition for the proposed QC-LDPC codes to have girth at least 6 was also analyzed. Regular and irregular QC-LDPC codes with various lengths and rates can be constructed. Simulations show that the proposed QC-LDPC codes have less short cycles and better performance than existing progressive edge growth based QC-LDPC codes.
In this work, the construction of low density parity check codes (LDPCs) from circulant permutation matrices is investigated. It is shown that such codes can not have a Tanner graph representation with girth larger than 12, and a relatively loose necessary and sufficient condition for the code to have a girth of 6, 8, 10 or 12 is derived. These results suggest that families of LDPC codes with such girth values are relatively easy to obtain and consequently, additional parameters such as the minimum distance or the number of redundant check sums should be considered.