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IEEE Information Theory Workshop (ITW)

LDPC Codes Based on Berlekamp-Justesen Codes with Large Stopping Distances

Xin Quan Ge, Shu‐Tao Xia

Published Oct 1, 2006• 11 citations
Publisher

Abstract

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

DOI
10.1109/itw2.2006.323790

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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
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.
Low-density parity check (LDPC) codes are an important class of codes with many applications. Two algebraic methods for constructing regular LDPC codes are derived -- one based on nonprimitive narrow-sense BCH codes and the other directly based on cyclotomic cosets. The constructed codes have high rates and are free of cycles of length four; consequently, they can be decoded using standard iterative decoding algorithms. The exact dimension and bounds for the minimum distance and stopping distance are derived. These constructed codes can be used to derive quantum error-correcting codes.
In this letter, the stopping sets and stopping distance of finite geometry LDPC (FG-LDPC) codes are studied. It is known that FG-LDPC codes are majority-logic decodable and a lower bound on the minimum distance can be thus obtained. It is shown in this letter that this lower bound on the minimum distance of FG-LDPC codes is also a lower bound on the stopping distance of FG-LDPC codes, which implies that FG-LDPC codes have considerably large stopping distance. This may explain in one respect why some FG-LDPC codes perform well with iterative decoding in spite of having many cycles of length 4 in their Tanner graphs.
This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly low-density parity-check (LDPC) codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descendant codes are developed, including the characterization of the roots of the generator polynomial of a cyclic descendant code. The second part of the paper shows that cyclic and quasi-cyclic descendant LDPC codes can be derived from cyclic finite-geometry LDPC codes using the results developed in the first part of the paper. This enlarges the repertoire of cyclic LDPC codes. The third part of the paper analyzes the trapping set structure of regular LDPC codes whose parity-check matrices satisfy a certain constraint on their rows and columns. Several classes of finite-geometry and finite-field cyclic and quasi-cyclic LDPC codes with large minimum distances are shown to have no harmful trapping sets of size smaller than their minimum distances. Consequently, their error-floor performances are dominated by their minimum distances.
A new method using Hamming codes to construct base matrices of (J,K)-regular LDPC convolutional codes with large free distance is presented. By proper labeling the corresponding base matrices and tailbiting these parent convolutional codes to given lengths, a large set of quasi-cyclic (QC) (J,K)-regular LDPC block codes with large minimum distance is obtained. The corresponding Tanner graphs have girth up to 14. This new construction is compared with two previously known constructions of QC (J,K)-regular LDPC block codes with large minimum distance exceeding (J + 1)!. Applying all three constructions, new QC (J,K)-regular block LDPC codes with J = 3 or 4, shorter codeword lengths and/or better distance properties than those of previously known codes are presented.