An Addition Structure on Incidence Matrices of a $BIB$ Design

Abstract

For a balanced incomplete block (BIB) design, the following problem is considered: Find $s$ different incidence matrices of the BIB design such that (i) for $1\le t\le s-1$, sums of any $t$ different incidence matrices yield BIB designs and (ii) the sum of all $s$ different incidence matrices becomes a matrix all of whose elements are one. In this paper, we show general results and present four series of such BIB designs with examples of three other BIB designs.