Abstract
A quasi-cyclic (QC) low-density parity-check (LDPC) code is called type-II, if the maximum weight over all circulants appearing in the parity-check matrix has the value of two. On the basis of multiplicative subgroup analysis for the prime field, a novel algebraic approach for type-II QC-LDPC codes is proposed from Tanner’s method. For column weight of four, the new type-II codes possess girth at least six and include a subset with very small circulant sizes almost attaining the theoretical lower bound. The new approach can yield type-II codes with two times smaller circulant sizes, in comparison with the state-of-the-art method. To enhance the flexibility of circulant sizes, a generalized Chinese-remainder-theorem (gCRT) method is proposed as well for type-II codes. Simulation results show that combining gCRT with the proposed short code yields compound type-II codes with a very promising decoding performance and flexible circulant sizes.
Highlights
Type-II quasi-cyclic (QC) low-density parity-check (LDPC) codes have attracted increasing attention [1]–[10], owing to the merit of commonly possessing larger upper bounds on distance [5], in comparison with traditional QC-LDPC codes [11]
Inspired by Tanner’s method, a novel class of type-II QC-LDPC codes with girth at least six is proposed, which can yield a subset with nearly the shortest circulant sizes
A generalized Chinese remainder theorem (CRT) method is presented to enhance the flexibility of circulant sizes or equivalently the code lengths for type-II QC-LDPC codes
Summary
Type-II quasi-cyclic (QC) low-density parity-check (LDPC) codes have attracted increasing attention [1]–[10], owing to the merit of commonly possessing larger upper bounds on distance [5], in comparison with traditional QC-LDPC codes [11]. Type-II codes with small circulant sizes can be explicitly constructed by a couple of algebraic methods, for example, [3], [6], [7], [10]. Among these studies, the state-of-the-art method (appendix B, [10]) can yield a class of type-II codes (with column weight of four) possessing circulant sizes only about two times larger than the theoretical lower bound.
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