The two key components in the SP-construction of an LDPC code are an SP-base matrix Bsp and a replacement set R of sparse member matrices. To ensure that the Tanner graph of an SP-LDPC code has girth at least 6, it is, in general, required that the SP-base matrix Bsp satisfies the RC-constraint and the member matrices in the replacement set R satisfy both the RC- and the PW-RC-constraints. In this chapter, we present several algebraic constructions of RC-constrained SP-base matrices and replacement sets whose member matrices satisfy both the RC- and the PW-RC-constraints. More constructions of RC-constrained SP-base matrices and replacement sets will be presented in Chapters 7 to 11 and Appendix A.

RC-Constrained Base Matrices

The algebraic methods presented in [58, 35, 3, 101, 112, 64, 65, 50, 113, 114, 46, 70, 68] can be used to construct RC-constrained SP-base matrices. All these methods are based on finite geometries, finite fields, and combinatorial designs (such as balanced incomplete block designs (BIBDs) or Latin squares). In this section, we use the construction based on finite Euclidean geometries. In Chapter 7, we will present a very flexible and powerful construction based on finite fields.

Consider the two-dimensional Euclidean geometry (EG) over the field GF(q) [84, 74, 97] (see Appendix A), denoted by EG(2,q). This geometry consists of q2 + q lines, each consisting of q points. Among these lines, q + 1 lines pass through the origin of the geometry. Based on the q2 - 1 lines of EG(2,q) not passing through the origin, it is possible to construct a (q + 1) × (q + 1) array HEG of CPMs and ZMs of size (q - 1) × (q - 1) [46, 26] which satisfies the RC-constraint [58, 97, 46] (see Appendix A). This array HEG contains q + 1 ZMs which can be put on the main diagonal of the array. Since HEG satisfies the RC-constraint, any subarray of HEG also satisfies the RC-constraint and hence can be used as an SP-base matrix for constructing an SP-LDPC code. The Tanner graph of the SP-LDPC code constructed based on this SP-base matrix has girth at least 6.