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IEEE Transactions on Information Theory

LDPC Codes Based on the Space of Symmetric Matrices Over Finite Fields

Changli Ma, Qi Wang, Meng Zhao

Published Mar 28, 2017• 5 citations
Publisher

Abstract

In this paper, we present a new method for explicitly constructing regular low-density parity-check (LDPC) codes based on Sn(Fq), the space of n × n symmetric matrices over Fq. Using this method, we obtain two classes of binary LDPC codes, C(n, q) and CT (n, q), both of which have grith 8. Then, both the minimum distance and the stopping distance of each class are investigated. It is shown that the minimum distance and the stopping distance of CT(n, q) are both 2q. As for C(n, q), we determine the minimum distance and the stopping distance for some special cases and obtain some lower bounds for other cases.

DOI
10.1109/tit.2017.2685501

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LDPC Codes Based on the Space of Symmetric Matrices over Finite Fields

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In this paper, we present a new method for explicitly constructing regular low-density parity-check (LDPC) codes based on Sn(Fq), the space of n × n symmetric matrices over Fq. Using this method, we obtain two classes of binary LDPC codes, C(n, q) and CT (n, q), both of which have grith 8. Then, both the minimum distance and the stopping distance of each class are investigated. It is shown that the minimum distance and the stopping distance of CT(n, q) are both 2q. As for C(n, q), we determine the minimum distance and the stopping distance for some special cases and obtain some lower bounds for other cases.
In this paper, we present a new method for explicitly constructing regular low-density parity-check (LDPC) codes based on Sn(Fq)\mathbb{S}_{n}(\mathbb{F}_{q})Sn​(Fq​), the space of n×nn\times nn×n symmetric matrices over Fq\mathbb{F}_{q}Fq​. Using this method, we obtain two classes of binary LDPC codes, C(n,q)\cal{C}(n,q)C(n,q) and CT(n,q)\cal{C}^{T}(n,q)CT(n,q), both of which have grith 888. Then both the minimum distance and the stopping distance of each class are investigated. It is shown that the minimum distance and the stopping distance of CT(n,q)\cal{C}^{T}(n,q)CT(n,q) are both 2q2q2q. As for C(n,q)\cal{C}(n,q)C(n,q), we determine the minimum distance and the stopping distance for some special cases and obtain the lower bounds for other cases.
Based on the codewords of the [q,2,q-1] extended Reed-Solomon (RS) code over the finite field Fq, we can construct a regular binary γq×q2 matrix H(γ,q), where q is a power of 2 and γ≤q. The matrix H(γ,q) defines a regular low-density parity-check (LDPC) code C(γ,q), called a full-length RS-LDPC code. Using some analytical methods, we completely determine the values of s(H(4,q)), s(H(5,q)), and d(C(5,q)) in this letter, where s(H(γ,q)) and d(C(γ,q)) are the stopping distance of H(γ,q) and the minimum distance of C(γ,q), respectively.
Let q be a power of 2 and y ≤ q an integer. Based on the codewords of [q, 2, q - 1] extended Reed-Solomon (RS) code over the finite field Fq, we can construct a (γ, q)-regular low-density parity-check (LDPC) code, called a full-length RS-LDPC code and denoted by C(γ, q). In this letter, the minimum distance of these codes is investigated. For any given q and y <; q, an upper bound on d(C(γ, q)), the minimum distance of C(γ, q), is provided. Furthermore, we determine the values of d(C(γ, q)) for y = 2, 3, and 4, and present the closed-form expressions on the numbers of minimum-weight codewords in C(γ, q) for γ = 2 and 3.
This correspondence studies the performance of the iterative decoding of low-density parity-check (LDPC) code ensembles that have linear typical minimum distance and stopping set size. We first obtain a lower bound on the achievable rates of these ensembles over memoryless binary-input output-symmetric channels. We improve this bound for the binary erasure channel. We also introduce a method to construct the codes meeting the lower bound for the binary erasure channel. Then, we give upper bounds on the rate of LDPC codes with linear minimum distance when their right degree distribution is fixed. We compare these bounds to the previously derived upper bounds on the rate when there is no restriction on the code ensemble.