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Electronics Letters

Deterministic construction of girth‐eight (3, L ) QC‐LDPC codes from quadratic function

Guohua Zhang, Rong Xia Sun, Xinmei Wang

Published Apr 1, 2013• 4 citations
Publisher

Abstract

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.

DOI
10.1049/el.2013.0320

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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 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 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.
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.
This article presents a simple, less computational complexity method for constructing exponent matrix (3, K) having girth at least 8 of quasi-cyclic low-density parity-check (QC-LDPC) codes based on subtraction method. The construction of code deals with the generation of exponent matrix by three formulas. This method is flexible for any block-column length K. The simulations are shown in comparison with some existing appreciable work. The codes with girth 8 are constructed with circulant permutation matrix (CPM) size P ≥ max{a2, r, a2, r,….a2, k} + 1.
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.