A fast construction algorithm for the incidence matrices of a class of symmetric balanced incomplete block designs

The theory of symmetric balanced incomplete block designs (BIBDs) has been applied in many research areas such as colored graphs, visual cryptography, distributed systems, communication networks, etc. In this paper, an explicit formula for a class of symmetric BIBDs is presented. Based on this formula, an efficient algorithm for constructing the incidence matrix of the design is developed. The incidence matrix contains all essential information of the design. The computational costs of the algorithm are O(v) which are superior to those of O(v(2)) or O(vrootv) by the conventional methods, where v is the number of objects or blocks.