Construction and Encoding of QC-LDPC Codes Using Group Rings

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 the 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 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.