4-Cycle Free LDPC Codes Based on Difference Sets

Difference sets are used to give a novel method for constructing structured regular quasi-cyclic low-density parity-check codes. Let F_q be a field with q elements and assume that D={d₁,..., d_k} is a (v,k,1)-difference set for Z_v with d₁<;d₂<;...<;d_k. Depending on v=q-1, v≥ 2d_k or v≥ 2d_k-1, three code construction methods are given that produce regular 4-cycle free codes, that is codes with girth at least 6. Simulation results show that the constructed codes perform well over the AWGN channel with an iterative sum-product message-passing decoding algorithm.