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IEEE Communications Letters

A Random Local Matroid Search Algorithm to Construct Good Rate $1/p$ Systematic Binary Quasi-Cyclic Codes

Guangfu Wu, Yong Li, Shuiping Zhang, Jiguang He

Published Feb 9, 2015• 1 citation
Publisher

Abstract

In this letter, a random local matroid search algorithm is proposed to construct good rate 1/p systematic QC codes based on the relationship between the generator matrix and the minimum distance when the length of the information vector is k ≥ 15 . More than twenty new QC codes are obtained and their corresponding generator matrices are revealed, which extends previous results. Nine of these new codes have larger minimum distances than the reported ones in the previous literature.

DOI
10.1109/lcomm.2015.2401572

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A Random Local Matroid Search Algorithm to Construct Good Rate 1/p1/p1/p Systematic Binary Quasi-Cyclic Codes
In this letter, a random local matroid search algorithm is proposed to construct good rate 1/p systematic QC codes based on the relationship between the generator matrix and the minimum distance when the length of the information vector is k ≥ 15 . More than twenty new QC codes are obtained and their corresponding generator matrices are revealed, which extends previous results. Nine of these new codes have larger minimum distances than the reported ones in the previous literature.
It has been well-known that the class of quasi-cyclic (QC) codes contain many good codes. In this paper, a method to conduct a computer search for binary 222-generator QC codes is presented, and a large number of good 222-generator QC codes have been obtained. 555 new binary QC codes that improve the lower bounds on minimum distance are presented. Furthermore, with new 222-generator QC codes and Construction X, 222 new improved binary linear codes are obtained. With the standard construction techniques, another 161616 new binary linear codes that improve the lower bound on the minimum distance have also been obtained. Received: 9 November 2016 Accepted: 22 November 2017
In this paper we present a method to search q circulant matrices; the concatenation of these circulant matrices with circulant identity matrix generates quasi-cyclic codes with high various code rate q/(q+1) (q an integer). This method searches circulant matrices in order to find the good quasi-cyclic code (QCC) having the largest minimum distance. A modified simulated annealing algorithm is used as an evaluator tool of the minimum distance of the obtained QCC codes. Based on this method we found 16 good quasi-cyclic codes with rates (1/2, 2/3 and 3/4), their estimated minimum distance reaches the lower bounds of codes considered to be the better linear block codes in Brouwer’s database.
The existing construction of Quasi-Cyclic low-density parity-check (short for QC-LDPC) codes has not considered the problems of small stopping sets and small girth and small minimum code weight together, while their existences will lead to the BER performance of QC-LDPC codes to be much poorer than that of randomly constructed LDPC codes even decoding failure. To solve the problem, some theorems of the QC-LDPC codes without small stopping sets and small girth were proposed first. Then good QC-LDPC codes without small stopping sets and small girth were designed based on the row vectors of generator matrix. The small minimum code weight also can be prevented due to the new algorithm based on the row vectors of generator matrix, which is the new way to choose the shifted factors to construct the good QC-LDPC codes for the given prime. The effectiveness and the practicability of the algorithm are demonstrated by the simulation results. It is significant for us to analyze and optimize the design of LDPC codes.
In this paper a new low complexity method for constructing binary quasi-cyclic low-density parity-check (QC-LDPC) codes is introduced. In the proposed method, each block-column of the parity check matrix H is made by a circulant matrix in a way that the associated Tanner graph is free of cycle four. Each circulant matrix in H is made by a generator column. The generator columns should be selected in a way that each associated circulant matrix and every two distinct circulant matrices are free of cycle four. The generator columns are made by row distance sets. An algorithm for generating distance sets and obtaining circulant matrices with columns of weight three is presented separately. Simplicity of construction and having a good flexible family of quasi cyclic LDPC codes both in rate and length are the main properties of the proposed method. The performance of the proposed codes is compared with that of the random-like and Array LDPC codes over an AWGN channel. Simulation results show that from the performance perspective, the constructed codes are competitive with random-like and Array LDPC codes.
To obtain efficient and pragmatic channel codes high signal-to-noise ratios (SNRs), an optimized construction of short and high rate protograph quasi-cyclic (QC) low-density parity-check (LDPC) codes is presented. Based on an optimized protograph template, the code framework is firstly produced by the extensions of the variable nodes. By enlarging the dimension of the sub-matrices related to the protograph framework, the base quasi-cyclic (QC) matrix is generated with required code rate and length. Then, the elements in the base matrix are split with even smaller square sub-matrices of the same row and column weights. In this procedure, a progressive edge growth (PEG) algorithm is employed to find the optimized positions of the QC submatrices, which brings larger girth for better code performance. Also the circulant offsets in all QC sub-matrices are optimally searched by a QC improved approximated cycle extrinsic message degree (QCI-ACE) algorithm, which improves the relationship of the unavoidable loops in the code's Tanner graph and thus it improves the error floor. Simulation results show that the codes produced by the proposed method demonstrate quite good bit error rate (BER) performance. In addition, they exhibit good properties of high spectrum efficiency by the high code rate, as well as the low complexity by the short code length. Therefore, the proposed code construction can be efficiently applied in the design of short and high rate protograph QC-LDPC codes with high rate and rational complexity.