Hengzhou Xu

Construction of Quasi-Cyclic LDPC Codes via Masking With Successive Cycle Elimination

In this letter, we study the construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes by array dispersion and masking. We first propose a class of arrays of circulant permutation matrices (CPMs) of arbitrary size obtained by dispersion. Based on this class of arrays of CPMs, a class of QC-LDPC codes is constructed. For these resulting codes, we analyze their cycle structure and derive the numbers, which indicate how many cycles pass through each CPM in their parity-check matrices. Following the indication numbers with the order from largest to smallest, we replace the corresponding CPMs with zero matrices (ZMs) of the same size one by one until a targeted masking termination condition is satisfied. Then, the resultant arrays of CPMs and ZMs give a large class of QC-LDPC codes. Numerical results show that the codes constructed perform well over the additive white Gaussian noise (AWGN) channel when decoded with the sum-product algorithm.

(3, L) quasi-cyclic LDPC codes: Simplified exhaustive search and designs

There exist lots of (3, L)-regular quasi-cyclic (QC) LDPC codes constructed from finite fields, protographs, array codes, and computer search under some design rules. For a given code length and rate, how to select the best one from these codes is considerable. In this paper, we study (3, L)-regular QC LDPC codes from the perspective of graph isomorphism, and non-isomorphic (3, L)-regular QC LDPC codes are determined. By analyzing the cycle structures of the resulting non-isomorphic codes, an efficient algorithm for counting cycles is presented. Also proposed is a simplified exhaustive search of non-isomorphic (3, L)-regular QC LDPC codes free of cycles of length less than g0, where g0 is the estimated optimal girth value for a given code length. Based on these two algorithms, we can easily construct a (3, L)-regular QC LDPC code with optimized cycle distribution for a given L and code length. Numerical results show that the constructed codes have better performance under the iterative decoding algorithms.

Construction of LDPC codes based on resolvable group divisible designs

In this paper, we present a method for constructing LDPC codes from resolvable group divisible designs (RGDDs). According to the incidence matrices (or their submatrices) of RGDDs, a class of regular LDPC codes can be constructed. Since the parity-check matrix of the resulting code satisfies the row-column constraint, the girth of the proposed codes is at least six. Furthermore, based on a resolvable group divisible design, we can obtain a sequence of LDPC codes with various lengths and rates, which perform well over the AWGN channel with iterative decoding and have low error floors.

Superposition Construction of

In this letter, we consider the construction of q-ary low-density parity-check (LDPC) codes by jointly optimizing the girth and number of shortest cycles in the frame of superposition. We begin with constructing binary quasi-cyclic LDPC codes by superposition. By replacing the nonzero entries of the parity-check matrices of the resulting binary codes with the nonzero elements of GF(q), a class of q-ary LDPC codes is obtained. Furthermore, we analyze the cycle structure of the constructed codes for a given degree distribution and code length, and present a search algorithm for finding q-ary LDPC codes with few cycles of length g, where g is the optimized girth value. Simulation results show that the proposed codes perform well under the iterative decoding algorithms.

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In this paper, we study the rank of matrices over and propose two construction methods for algebraic-based nonbinary LDPC codes from an existing LDPC code, referred to as the original code. By multiplying all elements of each column of the binary parity-check matrix H corresponding to the original code with the same nonzero element of any field, the first class of nonbinary LDPC codes with flexible field order is proposed. The second method is to replace the nonzero elements of some columns in H with different nonzero field elements in a given field, and then another class of nonbinary LDPC codes with various rates is obtained. Simulation results show that the proposed nonbinary LDPC codes perform well over the AWGN channel with the iterative decoding algorithms.

-Ary LDPC Codes by Jointly Optimizing Girth and Number of Shortest Cycles

This paper presents a family of rate-compatible (RC) globally coupled low-density parity-check (GC-LDPC) codes, which is constructed by combining algebraic construction method and graph extension. Specifically, the highest rate code is constructed using the algebraic method and the codes of lower rates are formed by successively extending the graph of the higher rate codes. The proposed rate-compatible codes provide more flexibility in code rate and guarantee the structural property of algebraic construction. It is confirmed, by numerical simulations over the AWGN channel, that the proposed codes have better performances than their counterpart GC-LDPC codes formed by the classical method and exhibit an approximately uniform gap to the capacity over a wide range of rates. Furthermore, a modified two-phase local/global iterative decoding scheme for GC-LDPC codes is proposed. Numerical results show that the proposed decoding scheme can reduce the unnecessary cost of local decoder at low and moderate SNRs, without any increase in the number of decoding iterations in the global decoder at high SNRs.

Algebraic-based nonbinary ldpc codes with flexible field orders and code rates

Dear editor, Nonbinary low-density parity-check (NBLDPC) codes have the ability of approaching capacity when decoded iteratively using a probabilistic decoding algorithm. With the increase of the finite field size q, NBLDPC codes over GF(q) have much better performance under iterative decoding for a constant code length. For q sufficiently large, there is no significant performance improvement while increasing q, moreover the column weights of the parity-check matrices of the best codes tend to 2. As an important class of NBLDPC codes, (2, ρ)-regular NBLDPC codes over GF(q) (q > 64), whose parity-check matrices have row weight ρ and column weight 2, perform well over various channels. This class of NBLDPC codes, so-called NBLDPC cycle codes, has attracted much attention [1–3]. Among these works, all designed codes have good performance. For a given code rate and code length, it is of great interest to study which one of them has the best error performance.

Construction and Decoding of Rate-Compatible Globally Coupled LDPC Codes

This paper proposes a two-stage decoding algorithm (called BP-LED) for short nonbinary low-density parity-check (LDPC) codes. It consists of the classical belief propagation (BP) decoder and a list erasure decoder (LED). Simulation results show that, for the (16, 8) LDPC code over GF(256) in the CCSDS standard, our proposed BP-LED algorithm can achieve a coding gain of 0.6 dB with respect to the FFT-QSPA. By choosing the parameters suitable for the proposed algorithm, it has a negligible performance loss with lower decoding complexity compared with the BP-MRB (most reliable basis) algorithm in [10].

Nonbinary LDPC cycle codes: efficient search, design, and code optimization

Quasi-cyclic (QC) LDPC codes whose parity-check matrices have diagonal structure play an important role in channel coding of 5G communications. In this paper, we study an algebraic-based method for constructing QC LDPC codes with diagonal structure of parity-check matrices. We first analyze the cycle structure of this class of QC LDPC codes and then divide the diagonal parity-check matrix into two parts, i.e., the diagonal matrix and the non-diagonal matrix. By employing the masking technique, we design the non-diagonal matrix based on prime field and QC LDPC codes with diagonal structure of parity-check matrices are proposed. Numerical results show that the constructed QC LDPC codes perform much better than the WiMAX-LDPC codes.