Generalized quasi-cyclic low-density parity-check codes based on finite geometries

Abstract

In this study, we proved that several promising classes of codes based on finite geometries cannot be classified as quasi-cyclic (QC) codes but should be included in broader generalized quasi-cyclic (GQC) codes. Further, we proposed an algorithm (transpose algorithm) for the computation of the Grobner bases from the parity check matrices of GQC codes. Because of the GQC structure of such codes, they can be encoded systematically using Grobner bases and their encoder can be implemented using simple feedback-shift registers. In order to demonstrate the efficiency of our encoder, we proved that the number of circuit elements in the encoder architecture is proportional to the code length for finite geometry (FG) LDPC codes. For codes constructed using points and lines of finite geometries, the hardware complexity of the serial-in serial-out encoder architecture of the codes is linear order O(n). To encode a binary codeword of length n, less than 2n adder and 3n memory elements are required.