Low-Complexity Transformed Encoder Architectures for Quasi-Cyclic Nonbinary LDPC Codes Over Subfields

Abstract

Quasi-cyclic low-density parity-check (QC-LDPC) codes are adopted in many digital communication and storage systems. The encoding of these codes is traditionally done by multiplying the message vector with a generator matrix consisting of dense circulant submatrices. To reduce the encoder complexity, this paper introduces two schemes making use of finite Fourier transform. We focus on QC-LDPC codes whose circulant submatrices are of dimension $(2^{r}-1)\times (2^{r}-1)$ and the entries are elements of GF $(2^{p})$ , where $p$ divides $r$ , and hence, GF $(2^{p})$ is a subfield of GF $(2^{r})$ . These cover a broad range of codes, and binary LDPC codes are a special case. Making use of conjugacy constraints, low-complexity architectures are developed for finite Fourier and inverse transforms over subfields in this paper. In addition, composite field arithmetic is exploited to eliminate the computations associated with message mapping and reduce the complexity of Fourier transform. For a (2016, 1074) nonbinary QC-LDPC code whose generator matrix consists of circulants of dimension $63 \times 63$ with GF $(2^{2})$ entries, the proposed encoders achieve 22% area reduction compared with the conventional encoders without sacrificing the throughput.