On the existence of balanced incomplete block designs with nested rows and columns

Abstract

A balanced incomplete block design with nested rows and columns, denoted by BIBRC ( v , k 1 , k 2 , λ ) , is a block design in which the plots of the blocks are arranged in arrays of size k 1 × k 2 such that λ = ( k 1 - 1 ) λ R ( i , j ) + ( k 2 - 1 ) λ C ( i , j ) - λ E ( i , j ) is a constant not depending on the choice of points i, j, where λ R ( i , j ) , λ C ( i , j ) and λ E ( i , j ) are the number of times that the pair i, j occurs in the same row, column and elsewhere, respectively. In this paper we will give a construction of BIBRCs by utilizing the method of differences. Moreover, in the case when λ satisfies some conditions, we will prove that there exists a BIBRC ( q , k 1 , k 2 , λ ) if q is a sufficiently large prime power and λ ( q - 1 ) ≡ 0 ( mod k 1 k 2 ( k 1 - 1 ) ( k 2 - 1 ) ) holds.