Quasi-Cyclic LDPC Codes on Two Arbitrary Sets of a ... - IEEE Xplore

2014 IEEE International Symposium on Information Theory

Quasi-Cyclic LDPC Codes on Two Arbitrary Sets of a Finite Field Juane Li, Keke Liu, Shu Lin and Khaled Abdel-Ghaffar Department of Electrical and Computer Engineering, University of California, Davis, CA 95616, USA Email: {jueli, kkeliu, shulin, ghaffar}@ucdavis.edu Abstract—This paper presents a simple and flexible method for constructing QC-LDPC codes based on two arbitrary sets of a finite field. Based on this method, a high-rate, high-performance and very low error-floor QC-LDPC code is first constructed and then a class of rate-1/2 QC-LDPC codes whose Tanner graphs have girth 8 or larger is presented. Also presented is a reducedcomplexity iterative decoding algorithm for QC-LDPC codes.

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I NTRODUCTION

Among various types of LDPC codes, quasi-cyclic (QC) LDPC codes are commonly preferred in practical applications due to their advantages in encoding and decoding implementations. There are two major approaches in construction of QC-LDPC codes, graph-theoretic and algebraic approaches. QC-LDPC codes constructed algebraically using mathematical tools such as finite fields, finite geometries and combinatorial designs, in general, give better overall performances in terms of coding gain, rate of decoding convergence and error-floors. This paper is concerned with construction of QC-LDPC codes of very high rate and rate-1/2 and their decoding with reduced decoding complexity. II.

A G ENERAL C ONSTRUCTION

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QC-LDPC C ODES

Let α be a primitive element of GF(q). Then, the powers of α, α−∞ , 0, α0 = 1, α1 , α2 , . . ., αq−2 , give all the q elements of GF(q). Let L = {−∞, 0, 1, . . . , q − 2}. For 1 ≤ m, n ≤ q, let S1 = αi0 , αi1 , . . . , αim−1 and S2 = αj0 , αj1 , . . . , αjn−1 be two arbitrary sets of elements in GF(q) with ik and jl in L, i0 < i1 < . . . < im−1 and j0 < j1 < . . . < jn−1 . Form the following m × n matrix over GF(q):   B = αik + αjl 0≤k