Additive, Structural and Multiplicative Transformations for the Construction of Quasi-Cyclic LDPC matrices

Abstract : The construction of a Quasi-Cyclic Low Density Parity-Check (QC-LDPC) matrix is usually carried out in two steps. In the first step, a prototype matrix is defined according to certain criteria (size, girth, check and variable node degrees, etc.). The second step involves expansion of the prototype matrix. During this last phase, an integer value is assigned to each non-null position in the prototype matrix corresponding to the right-rotation of the identity matrix. The problem of determining these integer values is complex. State-of-the-art solutions use either some mathematical constructions to guarantee a given girth of the final QC-LDPC code or a random search of values until the target girth is satisfied. In this paper, we propose an alternative/complementary method that reduces the search space by defining large equivalence classes of topologically identical matrices through row and column permutations using additive, structural and multiplicative transformations. Selecting only a single element per equivalence class can reduce the search space by a few orders of magnitude. Then, we use the formalism of constraint programming to list the exhaustive sets of solutions for a given girth and a given expansion factor. An example is presented in all sections of the paper to illustrate the methodology.
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Alban Derrien, Emmanuel Boutillon, Audrey Cerqueus. Additive, Structural and Multiplicative Transformations for the Construction of Quasi-Cyclic LDPC matrices. IEEE Transactions on Communications, Institute of Electrical and Electronics Engineers, 2019, 67 (4), pp.2647-2659. ⟨10.1109/TCOMM.2018.2890251⟩. ⟨hal-01950474⟩

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