LDPC Codes Based on α – Resolvable BIB and Group Divisible Designs

Structured LDPC codes have been
constructed using balanced incomplete block (BIB) designs, resolvable BIB
designs, mutually orthogonal Latin rectangles, partial geometries, group
divisible designs, resolvable group divisible designs and finite geometries.
Here we have constructed LDPC codes from α
– resolvable BIB and Group divisible
designs. The sub–matrices of incidence matrix of such block design are used as
a parity – check matrix of the code which satisfy row – column constraint. Here
the girth of the proposed code is at least six and the corresponding LDPC code
(or Tanner graph) is free of 4– cycles.


Introduction
Low-density parity-check (LDPC) codes were introduced by Gallager (1962). In recent years, these codes have become strong competitor to Turbo codes for error control in many digital storage and communication systems where high reliability is required. Based on the methods of constructions, LDPC codes can be divided into two types: random codes and structured codes. Random LDPC codes are constructed by computer search while structured LDPC codes are constructed by algebraic and combinatorial methods. Despite the excellent error-correcting properties of some known random LDPC codes, they often have the high complexity. A large amount of information is necessary to specify the positions of the non-zero elements in the parity-check matrix. These complexity drawbacks of random LDPC codes can be overcome by structured LDPC codes. A coverage on the constructions of structured LDPC codes using balanced incomplete block (BIB) and resolvable BIB designs, mutually orthogonal Latin rectangles, partial geometries, group divisible designs, resolvable group divisible designs and finite geometries may be found in Kou  A binary (γ, ρ)regular LDPC code is defined as the null space of an m x n sparse parity-check matrix H with entries from {0, 1} having the following properties: 1) each row has ρ nonzero elements; 2) each column has γ nonzero elements; 3) ρ n and γ m. The code rate of a (γ, ρ)regular LDPC code is 1 − γ/ρ provided that the parity check matrix has full rank. The Tanner graph of an LDPC code is a bipartite graph in which two types of nodes are variable nodes (representing coded symbol variables) and check nodes (representing the local check-sum constraints) respectively. The size of the shortest cycle is called the girth of the Tanner graph (or LDPC code). The shortest cycles affect the decoding performance with iterative algorithm based on belief propagation, especially the 4cycles. In order to create an LDPC code free of 4cycle, a structural property is imposed on H: no two distinct columns (or two distinct rows) have more than one nonzero element in common. This constraint on H is known as the row-column constraint (or RC constraint), see Diao et al. (2013). The RC-constraint ensures that the girth of the LDPC codes generated by such H is at least six.
The aim of this paper is to construct LDPC codes free of four cycles from αresolvable BIB and Group divisible designs. A submatrix of incidence matrix of such block design is used as paritycheck matrix of the code which satisfies rowcolumn constraint. Hence the girth of the proposed code is at least six and the corresponding LDPC code (or Tanner graph) is free of 4cycles.

Preliminaries
Suppose b blocks of a block design D (v, b, r, k) can be divided into ⁄ classes, each of size ⁄ such that in each class of blocks every point of D is replicated α times. Then these t classes are known as αresolution (or parallel) classes and the design is called αresolvable design. When α=1 the design is said to be resolvable and the classes are called resolution classes.

1. Construction from α -resolvable BIB designs
Consider The following theorem may be found in Raghavarao (1988)

and 3resolvability follows from
Bose's first module theorem, see Kageyama and Mohan (1983).

2. Construction from α -resolvable Group divisible designs
Consider a Group divisible design with parameters: v, b, r, k, Let and be two distinct blocks of the design. Since any two elements occur in either one block or no block, we have | | and hence any two distinct columns of the incidence matrix N have at most one nonzero element common. That is, every pair of points of the design is contained in at most one block and also every pair of blocks contains at most one point. Hence no two rows (or columns) in incidence matrix N have more than one position where they both have 1-component in common. If we consider N as parity-check matrix then it satisfies the RCconstraint and the constructed LDPC code (or Tanner graph) is free of 4cycles. Also each column and row sum of N is k and r respectively. Hence the LDPC code is (k, r)regular. Xu et al. (2015) constructed LDPC codes from resolvable GD designs. If (1) represents αresolvable GD design then we can construct LDPC codes as discussed in Section 3. 1.
Theorem 2 [Banerjee and Kageyama (1996)]: The existence of αresolvable GD design with parameters: implies the existence of another 2αresolvable GD design with parameters: Using Theorem 2 repeatedly to αresolvable GD design, we obtain following series:

Remark 2:
Resolvable solutions of SR31, SR32, SR33 and a 2resolvable solution of R88 will also yield series of GD designs which are applicable in the construction of LDPC codes.