In this chapter, we first present some definitions and concepts of matrices which are relevant for later developments of various algebraic-based and graph-theoretic-based constructions of LDPC codes. Then, we give a brief presentation of LDPC codes and characterize their fundamental structural properties that affect their error performance over the AWGNC and the BEC. Good coverage of the fundamentals of LDPC codes and the major iterative algorithms for decoding these codes can be found in [74, 97, 95].
Matrices and Matrix Dispersions of Finite Field Elements
Let GF(q) be a finite field with q elements, where q is a power of a prime . GF(q) is called a NB field if q ≠ 2 and a binary field if q = 2. A matrix A is said to be regular if every row has the same weight, called the row weight, every column has the same weight, called the column weight, and the row weight equals the column weight. If the row weight (or column weight) of a regular matrix A is w, we say that A has weight w and call it a weight-w regular matrix. For w ≠ 0, a weight-w regular matrix must be a square matrix. A binary weight-1 regular matrix is a permutation matrix (PM). It is clear that a weight-w regular matrix is the sum of w disjoint PMs. More generally, a rectangular matrix with constant column weight wc and constant row weight wr is said to be (wc, wr ) -regular.
A square matrix over GF(q) is called a circulant if every row of the matrix is the cyclic-shift of the row above it one place to the right, and the top row is the cyclic-shift of the last row one place to the right. It is clear that a circulant is a regular matrix whose weight equals the weight of its top row. The top row of a circulant is called the generator of the circulant. A binary circulant of weight 1 is a binary circulant permutation matrix (CPM). It is clear that a weight-w circulant is the sum of w disjoint CPMs.