Group-theoretic analysis of cayley-graph-based cycle gf(2^p) codes

Abstract

Abstract-Using group theory, we analyze cycle GF(2 p ) codes that use Cayley graphs as their associated graphs. First, we show that through row and column permutations the parity check matrix H can be put in a concatenation form of row-permuted block-diagonal matrices. Encoding utilizing this form can be performed in linear time and in parallel. Second, we derive a rule to determine the nonzero entries of H and present determinate and semi-determinate codes. Our simulations show that the determinate and semi-determinate codes have better performance than codes with randomly generated nonzero entries for GF(16) and GF(64), and have similar performance for GF(256). The constructed determinate and semi-determinate codes over GF(64) and GF(256) can outperform the binary irregular counterparts of the same block lengths. One distinct advantage for determinate and semi-determinate codes is that they greatly reduce the storage cost of H for decoding. The results in this correspondence are appealing for the implementation of efficient encoders and decoders for this class of promising LDPC codes, especially when the block length is large.

Structure, property, and design of nonbinary regular cycle codes

“…• First, encoding of nonbinary regular cycle codes can be done in linear time in parallel similar to [13]. This provides a lot of flexibility in the implementation of efficient encoders which is quite desirable especially when the codeword length is large.…”

“…The design of nonbinary regular cycle codes consists of code structure design and selection of nonzero entries of H. On the code structure design, one can design the code based on known graphs with large girth, such as the Ramanujan and Cayley graphs as done in [6], [13]. Or, one can rely on computer search based algorithms, such as the well-known progressive edge-growth (PEG) algorithm [14].…”

“…Or, one can rely on computer search based algorithms, such as the well-known progressive edge-growth (PEG) algorithm [14]. The research on the selection of nonzero entries has been widely pursued [13], [15]- [17]. In this paper, we propose a novel method that amounts to designing directly the interleavers in the equivalent structure of H developed herein.…”

“…Some of the structure and property results have been presented in our earlier work [13] for a special class of regular cycles codes constructed from Cayley graphs. The distinctions of this paper from [13] are as follows: i) the group-theoretic analysis adopted in [13] works only for codes based on Cayley graphs, hence the coverage of [13] is quite limited.…”

“…The distinctions of this paper from [13] are as follows: i) the group-theoretic analysis adopted in [13] works only for codes based on Cayley graphs, hence the coverage of [13] is quite limited. In this paper, we rely on graph-theoretic analysis, and the obtained results are applicable to a general regular cycle code; and ii) the results on the sequential BP decoding with parallel processing and the structure design of regular cycle codes are not available in [13]. In addition, this paper provides extensive simulation results on the code performance, with some of them on very-high-rate codes.…”