Deterministic construction of girth-eight (3,L) QC-LDPC codes from quadratic function

For any row weight L and any circulant permutation matrix size P > (L - 1)² + b₀(L - 1) (b₀ = L - 2 + mod(L,2)), a class of (3,L) quasi-cyclic (QC) low-density parity-check (LDPC) codes is explicitly constructed with girth eight, based on the quadratic function f(x) = x² + bx and its derivative, where b is either of the two integers {b₀, - b₀ - 2(L - 2)}. Compared with a similar construction from the so-called difference sequence, the proposed construction is not only much more simple in concept, but also completely deterministic in the sense that no computer search or computing is required. Simulation results show that the new codes significantly outperform the girth-eight codes constructed by the earliest-sequence-based method.