Title: Construction and Encoding of QC-LDPC Codes Using Group Rings,
Abstract: Quasi-cyclic (QC) low-density parity-check (LDPC) codes which are known as
QC-LDPC codes, have many applications due to their simple encoding
implementation by means of cyclic shift registers. In this paper, we construct
QC-LDPC codes from group rings. A group ring is a free module (at the same time
a ring) constructed in a natural way from any given ring and any given group.
We present a structure based on the elements of a group ring for constructing
QC-LDPC codes. Some of the previously addressed methods for constructing
QC-LDPC codes based on finite fields are special cases of the proposed
construction method. The constructed QC-LDPC codes perform very well over the
additive white Gaussian noise (AWGN) channel with iterative decoding in terms
of bit-error probability and block-error probability. Simulation results
demonstrate that the proposed codes have competitive performance in comparison
with the similar existing LDPC codes. Finally, we propose a new encoding method
for the proposed group ring based QC-LDPC codes that can be implemented faster
than the current encoding methods. The encoding complexity of the proposed
method is analyzed mathematically, and indicates a significate reduction in the
required number of operations, even when compared to the available efficient
encoding methods that have linear time and space complexities.