Two algebraic methods for systematic construction of structured regular and irregular low-density parity-check (LDPC) codes with girth of at least six and good minimum distances are presented. These two methods are based on geometry decomposition and a masking technique. Numerical results show that the codes constructed by these methods perform close to the Shannon limit and as well as random-like LDPC codes. Furthermore, they have low error floors and their iterative decoding converges very fast. The masking technique greatly simplifies the random-like construction of irregular LDPC codes designed on the basis of the degree distributions of their code graphs

This paper presents a superposition method for constructing low-density parity-check (LDPC) codes. Several classes of structured LDPC codes are constructed. Codes in these classes perform well with iterative decoding, and their Tanner graphs have girth at least six.

New algebraic methods for constructing codes based on hyperplanes of two different dimensions in finite geometries are presented. The new construction methods result in a class of multistep majority-logic decodable codes and three classes of low-density parity-check (LDPC) codes. Decoding methods for the class of majority-logic decodable codes, and a class of codes that perform well with iterative decoding in spite of having many cycles of length 4 in their Tanner graphs, are presented. Most of the codes constructed can be either put in cyclic or quasi-cyclic form and hence their encoding can be implemented with linear shift registers.

This letter presents two classes of quasi-cyclic low-density parity-check codes that perform close to the Shannon limit.

This correspondence presents a method for constructing structured regular low-density parity-check (LDPC) codes based on a special type of combinatoric designs, known as balance incomplete block designs. Codes constructed by this method have girths at least 6 and they perform well with iterative decoding. Furthermore, several classes of these codes are quasi-cyclic and hence their encoding can be implemented with simple feedback shift registers.

This correspondence presents three algebraic methods for constructing low-density parity-check (LDPC) codes. These methods are based on the structural properties of finite geometries. The first method gives a class of Gallager codes and a class of complementary Gallager codes. The second method results in two classes of circulant-LDPC codes, one in cyclic form and the other in quasi-cyclic form. The third method is a two-step hybrid method. Codes in these classes have a wide range of rates and minimum distances, and they perform well with iterative decoding.

The paper presents two classes of quasi-cyclic low-density parity-check (LDPC) codes which perform close to the Shannon limit. The construction of these codes is based on decomposition of circulant matrices constructed from finite geometries.

The notion of product block codes, whose generator matrix is the tensor product of the constituent generator matrices, is well established. Typically, they have good performance and a decoding algorithm with complexity on the order of the complexity of the decoding algorithms of the constituent codes. There has been much recent attention on the construction of bipartite graphs for low density parity check codes whose parity check matrices are the incidence matrices of right versus left vertices of the graph. The relation of the properties of the incidence matrix to code performance is difficult to establish precisely, although some guidelines are available. Two types of incidence matrix constructions are given here that show promise. In the first instance, a combinatorial construction is given, and, secondly, two types of graph products are considered for their application to LDPC codes.