A Construction Technique for Group Divisible (v-1,k,0,1) Partially Balanced Incomplete Block Designs (PBIBDs)

Group Divisible PBIBDs are important combinatorial structures with diverse applications. In this paper, we provided a construction technique for Group Divisible (v-1,k,0,1) PBIBDs. This was achieved by using techniques described in literature to construct Nim addition tables of order 2n, 2≤n≤5 and (k2,b,r,k,1)Resolvable BIBDs respectively. A “block cutting” procedure was thereafter used to generate corresponding Group Divisible (v-1,k,0,1) PBIBDs from the (k2,b,r,k,1)Resolvable BIBDs. These procedures were streamlined and implemented in MATLAB. The generated designs are regular with parameters(15,15,4,4,5,3,0,1);(63,63,8,8,9,7,0,1);(255,255,16,16,17,15,0,1) and (1023,1023,32,32,33,31,0,1). The MATLAB codes written are useful for generating the blocks of the designs which can be easily adapted and utilized in other relevant studies. Also, we have been able to establish a link between the game of Nim and Group Divisible (v-1,k,0,1) PBIBDs.

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

<p>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 <i>α </i>–<b> </b>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. </p>

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

<p>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 <i>α </i>–<b> </b>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. </p>