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Electronics Letters

Construction of girth‐eight QC‐LDPC codes from arithmetic progression sequence with large column weight

Yi Zhang, Xinyu Da

Published Jul 23, 2015• 14 citations
Publisher

Abstract

For any row weight L , a novel class of (8, L ) quasi‐cyclic low‐density parity‐check (QC‐LDPC) codes is deterministically constructed with girth eight. Compared with a similar construction, the proposed (7, L ) and (8, L ) QC‐LDPC codes exist for any circulant permutation matrix of size P ≥ ( L − 1)( L 2 + 3 L /4 + 3/4) + 1 and P ≥ ( L − 1)( L 2 + 3 L /4 + 7/4) +1, which remarkably improve the existing bounds of ( L − 1)( L 2 + L ) +1 and ( L − 1)( L 2 + L + 1) +1, respectively. Moreover, for any column weight J , any row weight L and any common difference, a construction for ( J , L ) QC‐LDPC codes with girth eight is also proposed. This construction has high flexibility with respect to the design of code rate and code length. Simulation results show that the novel codes with moderate lengths perform well in the additive white Gaussian noise channel.

DOI
10.1049/el.2015.0389

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For any row weight L , a novel class of (8, L ) quasi‐cyclic low‐density parity‐check (QC‐LDPC) codes is deterministically constructed with girth eight. Compared with a similar construction, the proposed (7, L ) and (8, L ) QC‐LDPC codes exist for any circulant permutation matrix of size P ≥ ( L − 1)( L 2 + 3 L /4 + 3/4) + 1 and P ≥ ( L − 1)( L 2 + 3 L /4 + 7/4) +1, which remarkably improve the existing bounds of ( L − 1)( L 2 + L ) +1 and ( L − 1)( L 2 + L + 1) +1, respectively. Moreover, for any column weight J , any row weight L and any common difference, a construction for ( J , L ) QC‐LDPC codes with girth eight is also proposed. This construction has high flexibility with respect to the design of code rate and code length. Simulation results show that the novel codes with moderate lengths perform well in the additive white Gaussian noise channel.
For any row weight L, several novel classes of (J = 5, L) and (J = 6, L) quasi-cyclic LDPC codes are deterministically constructed with girth eight. From these results, it is proved that (5, L) QC-LDPC codes with girth eight exist for any circulant permutation matrix (CPM) size P ≥ (2L + 3)(L - 1) + 1, and that girth-eight (6, L) QC-LDPC codes exist for any P ≥ 2(L+5)(L-1)+1. The two novel bounds remarkably improve the existing bounds of L2(L-1) + 1 and (L2+1)(L-1)+1, respectively. Moreover, for any column weight J and any row weight L, a construction for (J,L) QC-LDPC codes with girth eight is also proposed. This is the first deterministic and systematic construction which can generate girth-eight QC-LDPC codes with J ≥ 7.
For any row weight L and any circulant permutation matrix size P > ( L − 1) 2 + b 0 ( L − 1) ( b 0 = L − 2 + mod( L ,2)), a class of (3, L ) quasi‐cyclic (QC) low‐density parity‐check (LDPC) codes is explicitly constructed with girth eight, based on the quadratic function f ( x ) = x 2 + bx and its derivative, where b is either of the two integers { b 0 , − b 0 − 2( L − 2)}. Compared with a similar construction from the so‐called difference sequence, the proposed construction is not only much more simple in concept, but also completely deterministic in the sense that no computer search or computing is required. Simulation results show that the new codes significantly outperform the girth‐eight codes constructed by the earliest‐sequence‐based method.
For any column weight J and any row weight L, a novel framework is proposed such that a girth-eight (J,L) quasi-cyclic low-density parity-check (QC-LDPC) code with any block length above a lower bound can be constructed via a simple inequality in terms of greatest common divisor (GCD). The main advantage is that the construction of a class of (J,L) girth-eight QC-LDPC codes is transformed into a rather simple task, searching for J integers satisfying the so-called GCD constraint for L. Combining the new method with masking matrices, a class of type-1 QC-LDPC codes is presented with girth at least eight. Simulation results show that the type-1 codes perform better than the random QC-LDPC codes and quadratic-congruence-based QC-LDPC codes for moderate block lengths and low code rates.
Quasi-cyclic low-density parity-check (QC-LDPC) codes are known to have algebraic structure, which suits hardware implementation. However, low girth makes their performances worse than random structured LDPC codes. Based on independent row-column mapping sequence, this work proposed a generalized method of constructing QC-LDPC codes with girth eight, which enlarges the class of girth eight QC-LDPC codes to a fairly general series. Simulation results show that the bit error rate of the constructed codes is close to that of progressive edge growth (PEG) based QC-LDPC codes under iterative decoding.
Recently, for each row weight KKK and column-weight JJJ, 3leJ<K3le J< K3leJ<K, several classes of (J,K)(J,K)(J,K) quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth 8 have been constructed explicitly such that their corresponding lower-bounds on the size of circulant permutation matrices (CPMs) have been considered small as possible. In this paper, for J=7J=7J=7, a class of (7,K)(7,K)(7,K) QC-LDPC codes with girth 8 is generated by an explicit method such that the lower-bounds of the constructed codes remarkably are better than the state-of-the-art bound (K−1)(K2+K)+1(K-1)(K^2+K)+1(K−1)(K2+K)+1.
A simple and efficient search method based on the coset concept is proposed to construct (3, L ) quasi‐cyclic low‐density parity‐check (QC‐LDPC) codes without small cycles. Compared with the random search method, the chance to find large‐girth codes is remarkably increased for the new method. Compared with Tanner's construction, the proposed method is much more flexible in the selection of row weight L . Some new codes are constructed with the shortest lengths compared with those constructed by the existing well‐known methods. Simulation results show that the novel codes with moderate length perform well in the additive white Gaussian noise channel.