Anti Quasi-Cyclic LDPC Codes

Low-density parity-check (LDPC) codes based on affine permutation matrices, APM-LDPC codes, have been attracted recently, because of some advantages rather than QCLDPC codes in minimum-distance, cycle distribution and errorrate performance. In this letter, circulant and anti-circulant permutation matrices are used to define a class of LDPC codes, called AQC-LDPC codes, which can be considered as an special case of APM-LDPC codes. In fact, each AQC-LDPC code can be verified by a sign matrix and a slope matrix which are helpful to show each cycle in the Tanner graph by a modular linear equation. For the normal sign matrix A, if -1 is an element of A, it is shown that the corresponding AQC-LDPC code has maximum-girth 8. Finally, two explicit constructions for AQC-LDPC codes with girths 6, 8 are presented which have some benefits rather than the explicitly constructed QC and APM LDPC codes in minimumdistance, cycle distributions and bit-error-rate performances.