CHAPTER 14 - Some Combinatorial Problems and Results in Fractional Replication

Abstract

This chapter focuses on the combinatorial problems and results in fractional replication. It presents the matrix D that is composed of n + 1 row vectors of size 1 x n, with each vector containing from zero to n ones with all elements not one being zero. Every row of D is uniquely distinct and the total number of ones, T, in D is equal to n + k, k = 0, 1, n2—n. Here t equals the smallest number of ones in any row vector of the matrix and r equals the number of row vectors with t ones. When there are r row vectors with t ones, the remaining n + l − r row vectors should contain n + k − rt ones. If matrices are formed by starting with r row vectors with t ones in each and if n + 1 − r >n + k − rt, then no matrix exists because the number of row vectors left to choose is greater than the number of ones left to distribute.