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IEEE Communications Letters

Automatic Verification of GCD Constraint for Construction of Girth-Eight QC-LDPC Codes

Guohua Zhang, Yi Fang, Yuanhua Liu

Published Jun 28, 2019• 11 citations
Publisher

Abstract

As a general method to yield constructions for quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth eight, the greatest-common-divisor (GCD) framework heavily relies on verifying a type of inequalities, referred to as GCD constraints. An algorithm is developed in this letter to automatically verify GCD constraints without conducting any manual analysis, by bounding from above the GCD of a fixed integer and an integer in the form of linear function. As an application of the algorithm, a set of novel constructions based on GCD framework is proposed. From these new constructions, four novel bounds on the size of circulant permutation matrices (CPMs) are formulated, such that girth-eight QC-LDPC codes always exist for any CPM size greater than or equal to the bounds. The new bound for column weight of 6 slightly improves the existing best one, and those for column weights from 7 to 9 significantly strengthen the state-of-the-art ones by decreasing from essentially a cubic or biquadratic power of row weight to a quadratic power of row weight.

DOI
10.1109/lcomm.2019.2925792

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IEEE Access•Jan 1, 2023•10 citations
As a general method to yield constructions for quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth eight, the greatest-common-divisor (GCD) framework heavily relies on verifying a type of inequalities, referred to as GCD constraints. An algorithm is developed in this letter to automatically verify GCD constraints without conducting any manual analysis, by bounding from above the GCD of a fixed integer and an integer in the form of linear function. As an application of the algorithm, a set of novel constructions based on GCD framework is proposed. From these new constructions, four novel bounds on the size of circulant permutation matrices (CPMs) are formulated, such that girth-eight QC-LDPC codes always exist for any CPM size greater than or equal to the bounds. The new bound for column weight of 6 slightly improves the existing best one, and those for column weights from 7 to 9 significantly strengthen the state-of-the-art ones by decreasing from essentially a cubic or biquadratic power of row weight to a quadratic power of row weight.
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.
The greatest-common-divisor (GCD) method is a general framework employing a set of simple inequalities (called GCD constraint) to guarantee girth eight for a class of (J,L) quasi-cyclic (QC) low-density parity-check (LDPC) codes. However, an important problem, i.e., whether the GCD constraint is necessary for this class of codes to have girth eight, remains open. In this letter, the question is answered affirmatively, following which a novel algorithm aiming to find the shortest codes with girth eight in such a class is proposed. Besides, a close connection is established between the GCD method and the base expansion method, which are both applicable for any J and any L.
Quasi-cyclic (QC) LDPC codes with large girths play a crucial role in several research and application fields, including channel coding, compressed sensing and distributed storage systems. A major challenge in respect of the code construction is how to obtain such codes with the shortest possible length (or equivalently, the smallest possible circulant size) using algebraic methods instead of search methods. The greatest-common-divisor (GCD) framework we previously proposed has algebraically constructed QC-LDPC codes with column weights of 5 and 6, very short lengths, and a girth of 8. By introducing the concept of a mirror sequence and adopting a new row-regrouping scheme, QC-LDPC codes with column weights of 7 and 8, very short lengths, and a girth of 8 are proposed for arbitrary row weights in this article via an algebraic manner under the GCD framework. Thanks to these novel algebraic methods, the lower bounds (for column weights 7 and 8) on consecutive circulant sizes are both improved by asymptotically about 20%, compared with the existing benchmarks. Furthermore, these new constructions can also offer circulant sizes asymptotically about 25% smaller than the novel bounds.
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.
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.
As a special case of a quasi-cyclic (QC) low-density parity-check (LDPC) code, a full-length row-multiplier (FLRM) QC-LDPC code is described by a compact exponent matrix based on two sequences of integers. The codes designed by a framework known as greatest-common-divisor (GCD) method, belong to a salient class of FLRM QC-LDPC codes, which can eliminate cycles of length up to six by carefully selecting a special sequence subject to a set of simple inequalities. However, the GCD method ensures the absence of these cycles only if circulant sizes are larger than a certain threshold. By combining the existing GCD method, novel sequences and a new analysis method (based on new lemmas of circulants and integers) for modulo equations, a group of novel FLRM QC-LDPC codes free of 4-cycles and 6-cycles are explicitly proposed for column weights from three to five in this paper, which possess circulant sizes much smaller than the forgoing threshold. Simulations show that the new FLRM QC-LDPC codes with shorter lengths perform almost the same as the existing FLRM QC-LDPC codes with longer lengths, and that the novel FLRM QC-LDPC codes noticeably outperform their counterparts with (nearly) identical lengths.