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

Using group theory, we analyze cycle GF(2) 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.