Combinatorially Designed LDPC Codes Using Zech Logarithms and Congruential Sequences

Abstract

We investigate a systematic construction of regular low density parity check (LDPC) codes based on (γρ^γ-1, ρ^γ, ρ, γ{0,1}) combinatorial designs. The proposed (γ,ρ) regular LDPC ensemble has rate ${\left( {1 - \tfrac{1}{\rho }} \right)^\gamma }$ girth ≥2^γ+1, and exists for all γ≥2. The codes are a subset of Gallager’s random ensemble, but contains a good combination of structure and (pseudo)randomness. In particular, the simple case of γ=2 results in a class of codes that are high-rate, systematic, quasi-cyclic, linear-time encodable and decodable, and free of length-4 and length-6 cycles. Analysis on distance spectrum shows that they are better than the Gallager ensemble of the same parameters. Simulation of the proposed codes on intersymbol interference channels show that they perform comparably to random LDPC codes. Unlike random codes, the proposed structured LDPC codes can lend themselves to a low-complexity implementation for high-speed applications.