Computing of Blocks of Some Combinatorial Designs for Applications

Abstract

Combinatorial block designs are used in various fields of technology; recently, they have been implemented, for example, as models of computer systems and key distribution schemes. They are defined on the basis of some finite set as satisfying certain conditions the set of its subsets called blocks. In this chapter, we study combinatorial block designs whose order is a prime number or a prime power. The tasks sufficient for the practical use of such block designs are considered: determining the composition of a particular block by its number and determining the numbers of blocks containing given element. It is shown that for their solution, a memory volume linear in the design order is used. Solutions are given with a certain numbering of elements and blocks. The sets on which the designs are defined are initial nonnegative, the blocks are their subsets. Computations are performed by mapping this set into a vector space of a certain dimension over the prime field or its extention. Four algorithms are given: Algorithm 1 and Algorithm 2 for constructing of projective plane and dual projective plane and Algorithms 3 and 4 for constructing of separate blocks of those projective planes.