Problem
Research questions and friction points this paper is trying to address.
QC-LDPC codes
girth
minimum distance
permutation matrices
iterative decoding
Low-Density Parity-Check Codes of High Girth from Permutation Polynomials
🤖 AI Summary
This study addresses the limitations in girth and minimum distance of quasi-cyclic low-density parity-check (QC-LDPC) codes caused by the commutativity of circulant permutation matrices. To overcome this, we propose a protograph code construction method based on non-commutative binomial permutation polynomials. By introducing higher-order permutation polynomials to construct the protograph, the proposed approach exploits their non-commutative properties to surpass the girth upper bound of conventional LDPC codes while preserving structural simplicity through iterative decoding algorithms. Simulation results demonstrate that the decoding performance of the proposed scheme significantly outperforms traditional QC-LDPC constructions based on affine permutation polynomials. This work provides a promising new avenue for the design of high-performance error-correcting codes.
📝 Abstract
Quasi-cyclic low-density parity-check (QC-LDPC) codes are widely used in modern communication standards due to their simple structure and efficient encoder and decoder implementations. However, the commutativity of the circulant permutation matrices used in their construction significantly limits their girth and minimum distance, affecting iterative decoding performance. In this paper, we investigate protograph-based codes constructed using permutation polynomials of higher degree, focusing on a closed family of non-commuting binomial permutation polynomials that enables these constructions to exceed both of these girth and minimum distance upper bounds while maintaining a simple representation. Simulation results demonstrate improved decoding performance compared to constructions based on affine permutation polynomials, which include QC-LDPC codes.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Quasi-cyclic LDPC codes
Permutation polynomials
High girth
Non-commuting binomials
Protograph-based codes
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