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Convolutional cylinder-type block-circulant cycle codes

Mohammad Gholami, Mehdi Samadieh

Published Jun 1, 2013• 1 citation
OpenAlex

Abstract

In this paper, we consider a class of column-weight two quasi-cyclic low-density paritycheck codes in which the girth can be large enough, as an arbitrary multiple of 8. Then we devote a convolutional form to these codes, such that their generator matrix can be obtained by elementary row and column operations on the parity-check matrix. Finally, we show that the free distance of the convolutional codes is equal to the minimum distance of their block counterparts.

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Problems of Information Transmission•Dec 1, 2009•12 citations
In this paper, we consider a class of column-weight two quasi-cyclic low-density paritycheck codes in which the girth can be large enough, as an arbitrary multiple of 8. Then we devote a convolutional form to these codes, such that their generator matrix can be obtained by elementary row and column operations on the parity-check matrix. Finally, we show that the free distance of the convolutional codes is equal to the minimum distance of their block counterparts.
A class of geometrically structured quasi-cyclic low-density parity-check codes with a cylinder structure graph and girth 12 is considered. The parity-check matrix of such a code consists of blocks of zeros and cir-culant permutation matrices. A class of convolutional codes is assigned to these codes. The convolutional form generator matrix of these codes is obtained by elementary row and column operations. It is shown that the free distance of the obtained convolutional codes is equal to the minimum distance of their bock code counterparts.
In this paper, a particular class of block-circulant low-density parity-check (BC-LDPC) codes referred to as cylinder-type BC-LDPC (CTBC-LDPC) codes is studied. We represent a cylinder-type parity-check matrix H by a graph called the block-structure graph of H and denoted by BSG(H). Using the properties of BSG(H) we show that CTBC matrices with column-weight two and girth an arbitrary multiple of 8 can be constructed, while for a CTBC matrix H with column-weight ℓ ≥ 3 this girth cannot exceed 12. An algorithm generating CTBC-LDPC codes of arbitrary possible girth is given. The algorithm produces a large group of codes with flexible rate, length and girth. From performance perspective over AWGN channel, the maximum girth-12 CTBC-LDPC codes with column-weight at least three generated by the given algorithm are at least as good as their random-like counterpart codes and outperform the codes of the same size constructed by successive-level-growth, progressive-edge-growth, and partition-and-shift methods.
Four classes of maximum-girth geometrically structured column-weight-two regular quasi-cyclic (QC) low-density parity-check (LDPC) codes are introduced. Two classes of these codes, referred to as Type-I and Type-II codes, are with row-weights 4 and 3, and maximum girths 16 and 24, respectively. The idea behind the construction of these two classes of codes, with rates at least 1/2 and 1/3, is slightly generalized to obtain two classes of variable-high-rate codes, referred to as Type-III1 and Type-III2 codes, with maximum girth 20 and 16, respectively. A low-complexity deterministic algorithm for constructing these four classes of codes is given. The algorithm generates maximum-girth Type-I and Type-II codes with almost arbitrary length eta not less than 216 and 243, respectively. The output of the algorithm substantially improves on some of the previously best known codes constructed using a randomized progressive edge-growth (RPEG) algorithm. For instance, we have rate-0.71 Type-III1 codes of lengths 308 and 728 with girths 10 and 12, respectively, versus the code lengths 385 and 840 obtained by the RPEG algorithm. Simulation results on AWGN channel confirm that, from BER performance perspective, the constructed LDPC codes are superior to the column-weight-two LDPC codes constructed by the previously reported methods. The generator matrix G(D) of the convolutional codes associated with Type-I and Type-II codes is given. The free distance dfreeof such a convolutional code is equal to the minimum distance of the corresponding QC block code.
A class of algebraically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes and their convolutional counterparts is presented. The QC codes are described by sparse parity-check matrices comprised of blocks of circulant matrices. The sparse parity-check representation allows for practical graph-based iterative message-passing decoding. Based on the algebraic structure, bounds on the girth and minimum distance of the codes are found, and several possible encoding techniques are described. The performance of the QC LDPC block codes compares favorably with that of randomly constructed LDPC codes for short to moderate block lengths. The performance of the LDPC convolutional codes is superior to that of the QC codes on which they are based; this performance is the limiting performance obtained by increasing the circulant size of the base QC code. Finally, a continuous decoding procedure for the LDPC convolutional codes is described.
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.
We consider regular block and convolutional LDPC codes determined by paritycheck matrices with rows of a fixed weight and columns of weight 2. Such codes can be described by graphs, and the minimum distance of a code coincides with the girth of the corresponding graph. We consider a description of such codes in the form of tail-biting convolutional codes. Long codes are constructed from short ones using the “voltage graph” method. On this way we construct new codes, find a compact description for many known optimal codes, and thus simplify the coding for such codes. We obtain an asymptotic lower bound on the girth of the corresponding graphs. We also present tables of codes.