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Journal of Algebraic Systems

(7,K) GIRTH-8 QC-LDPC CODES WITH AN EXPLICIT CONSTRUCTION

M. Majdzadeh, Mohammad Gholami, Gh. Raeisi

Published Jan 1, 2022• 2 citations
Publisher

Abstract

Recently, for each row weight $K$ and column-weight $J$, $3le J< K$, several classes of $(J,K)$ 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 $J=7$, a class of $(7,K)$ 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 $(K-1)(K^2+K)+1$.

DOI
10.22044/jas.2021.8911.1434

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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.
Recently, some attentions have been focused on a class of low-density parity-check codes from affine permutation matrices, called APM-LDPC codes, which have some advantages rather than quasi-cyclic low-density parity-check (QC-LDPC) codes in the minimum-distance, cycle distribution and error-rate performance. In this letter, first, some novel explicit constructions for exponent matrices of conventional APM-LDPC codes with girths 6, 8, and 10 are investigated, and for each given exponent matrix E corresponding to an APM-LDPC code with girth 2g , 3 ≤ g ≤ 5, a lower bound Q = Q(E,g) is obtained, such that APM-LDPC codes with exponent matrix E have girth at least 2g, for each (prime) APM size m ≥ Q. Then, regarding to reduce the lower bound, an improved bound IQ is obtained, which is significantly better than some lower bounds reported for QC-LDPC codes with explicit constructions. Simulation results show that the constructed APM-LDPC codes outperform QC-LDPC codes with the same girth.
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.
Recently, a class of low-density parity-check codes based on affine permutation matrices, called APM-LDPC codes, have been considered which have some advantages than quasi-cyclic (QC) LDPC codes in terms of minimum-distance, cycle distribution, and error-rate performance. Moreover, some explicit constructions for exponent matrices of conventional APM-LDPC codes with girth at least 6 have been investigated. In this paper, a class of 4-cycle free APM-LDPC codes is constructed by a new explicit method such that the constructed codes have better cycle distributions rather than the recently proposed APM codes with girth 6. As simulation results show, the constructed codes outperform PEG and random-like LDPC codes with the same rates and lengths.
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.