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A New Construction Method for Large Girth Quasi-Cyclic LDPC Codes with Optimized Lower Bound using Chinese Remainder Theorem

Ambar Bajpai, Gan Srirutchataboon, Piya Kovintavewat, Lunchakorn Wuttisittikulkij

Published Jun 29, 2016• 9 citations
Publisher
DOI
10.1007/s11277-016-3465-8

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For an arbitrary (3, L ) quasi-cyclic(QC) low-density parity-check (LDPC) code with girth at least ten, a tight lower bound of the consecutive lengths is presented. For an arbitrary length above the bound the corresponding LDPC code necessarily has a girth at least ten, and for the length equal to the bound, the resultant code inevitably has a girth smaller than ten. This new conclusion can be well applied to some important issues, such as the proofs of the existence of large girth QC-LDPC codes, the construction of large girth QC-LDPC codes based on the Chinese remainder theorem, as well as the construction of LDPC codes with the guaranteed error correction capability.
In this letter, we consider two problems associated with quasi-cyclic low-density parity-check (QC-LDPC) codes. The first is how to extend the code length of a QC-LDPC code without reducing the girth. The second is how to design a QCLDPC code with a prescribed girth easily. We deal with these two problems by using a combining method of QC-LDPC codes via the Chinese Remainder Theorem (CRT). Codes constructed with our proposed method have flexible code lengths, flexible code rates and large girth. Simulation results show that they perform very well with the iterative decoding.
In this paper we propose a method of constructing quasi-cyclic low-density parity-check (QC-LDPC) codes of large length by combining QC-LDPC codes of small length as their component codes, via the Chinese remainder theorem. The girth of the QC-LDPC codes obtained by the proposed method is always larger than or equal to that of each component code. By applying the method to array codes, we present a family of high-rate regular QC-LDPC codes with no 4-cycles. Simulation results show that they have almost the same performance as random regular LDPC codes.