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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.

Type-II QC-LDPC Codes From Multiplicative Subgroup of Prime Field