On the Nonexistence of Knut Vik Designs for all Even Orders

Abstract

A Kunt Vik design of order $n$ can be defined as an $n \times n$ array of elements, chosen from a set of $n$ elements (treatments) such that with respect to rows and columns the array is a Latin square and in addition each treatment appears once in each of the $n$ left and right diagonals. These designs are useful for eliminating sources of variation in four directions. This paper is concerned with the existence and nonexistence of these designs. Specifically, (i) it is shown that no such design exists for $n$ even, (ii) these designs exist for all odd orders except possibly for $n \equiv 0 (\mod 3)$, (iii) the Kronecker product of two Knut Vik designs in a Kunt Vik design and (iv) the concept of semi Knut Vik design is also defined and it is shown that while these designs do not exist for even orders, they exist for all odd orders.