On Advisability of Designing Short Length QC-LDPC Codes Using Perfect Difference Families

A simple and general definition of quasi cyclic low-density parity-check (QC-LDPC) codes which are constructed based on circulant permutation matrices (CPM) is proposed. As an special case of this definition, we first represent one type of so called combinatorially designed multiple-edge protograph codes. The code construction is mainly based on perfect difference families (PDF’s) and is called Construction 1. Secondly, using the proposed Construction 1 along with a technique named as column dispersion technique (CDT), we design several types of multiple-edge CPM-QC-LDPC codes (codes with Construction 2) in a wide range of rates, lengths, girths and minimum distances. Parameters of some of these codes are reported in tables. Also included in this paper are the multiplicities of short simple cycles of length up to 10 in Tanner graph of our constructed codes. Our experimental results for short to moderate length codes show that although minimum distances of codes play an important role in waterfall region, the higher the number of short simple cycles is, the better (sharper) the waterfall is. The performances of many codes such as WiMAX, PEG, array, MacKay, algebraic and combinatorial, and also, symmetrical codes have compared with our constructed codes. Based on our numerical results and regardless of how a code is constructed, those with higher number of short simple cycles and higher minimum distances, have better waterfalls. As, high number of short simple cycles cause error floor phenomenon, we show that our constructed codes based on applying CDT on PDF’s have a property that we can gradually increase their number of short simple cycles to outperform many of the codes in the literature whilst, preventing them to have very high number of short cycles and thus avoiding undesirable error floors. A. Tasdighi and M. R. Sadeghi are with the Department of Mathematics & Computer Science, Amirkabir University of Technology, Tehran, Iran (e-mails: [email protected], [email protected]) 2 Index Terms QC-LDPC codes, QC-LDPC code construction, Perfect difference families, Girth, Minimum distance, Inevitable cycle, Error floor