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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Further Results on the Minimum and Stopping Distances of Full-Length RS-LDPC Codes

Haiyang Liu, Hao Zhang, Lianrong Ma

Published Jan 1, 2017• 1 citation
Publisher

Abstract

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.

DOI
10.1587/transfun.e100.a.738

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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.
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.
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 work, we study the minimum/stopping distance of array low-density parity-check (LDPC) codes. An array LDPC code is a quasi-cyclic LDPC code specified by two integers q and m, where q is an odd prime and m <= q. In the literature, the minimum/stopping distance of these codes (denoted by d(q,m) and h(q,m), respectively) has been thoroughly studied for m <= 5. Both exact results, for small values of q and m, and general (i.e., independent of q) bounds have been established. For m=6, the best known minimum distance upper bound, derived by Mittelholzer (IEEE Int. Symp. Inf. Theory, Jun./Jul. 2002), is d(q,6) <= 32. In this work, we derive an improved upper bound of d(q,6) <= 20 and a new upper bound d(q,7) <= 24 by using the concept of a template support matrix of a codeword/stopping set. The bounds are tight with high probability in the sense that we have not been able to find codewords of strictly lower weight for several values of q using a minimum distance probabilistic algorithm. Finally, we provide new specific minimum/stopping distance results for m <= 7 and low-to-moderate values of q <= 79.
In this paper, we study the minimum/stopping distance of array low-density parity-check (LDPC) codes. An array LDPC code is a quasi-cyclic LDPC code specified by two integers q and m, where q is an odd prime and m q. In the literature, the minimum/stopping distance of these codes (denoted by d(q, m) and h(q, m), respectively) has been thoroughly studied for m 5. Both exact results, for small values of q and m, and general (i.e., independent of q) bounds have been established. For m = 6, the best known minimum distance upper bound, derived by Mittelholzer, is d(q, 6) 32. In this paper, we derive an improved upper bound of d(q, 6) 20 and a new upper bound d(q, 7) 24 by using the concept of a template support matrix of a codeword/stopping set. The bounds are tight with high probability in the sense that we have not been able to find codewords of strictly lower weight for several values of q using a minimum distance probabilistic algorithm. Finally, we provide new specific minimum/stopping distance results for m 7 and low-to-moderate values of q ≤79.
By employing a class of q-ary cyclic codes, i.e., Berlekamp-Justesen (B-J) codes, as base codes, we obtain two classes of structured LDPC codes by q-tuple and (q - 1)-tuple substitutions. These constructions can be viewed as generalizations of Reed-Solomon based LDPC codes proposed by Djurdjevic et. al. The B-J based LDPC codes, which could be regular or irregular according to the adjustment of three parameters, have much flexibility in choices of the code length, rate and minimum/stopping distance. Furthermore, the minimum distances and stopping distances of the B-J based LDPC codes are analyzed. We show that the stopping distance of a B-J based LDPC code is not smaller than the best known lower bound of its minimum distance. The B-J based LDPC codes perform well under iterative decoding and manifest low error-floors in simulations which could be explained in one respect by their large minimum distances and stopping distances