Existence of \(V(9,t) \) Vectors

A \( V(m,t) \) leads to \( m \) idempotent pairwise orthogonal Latin squares of order \( (m+1)t+1 \) with one common hole of order \( t \). \( V(m,t) \)’s can also be used to construct perfect Mendelsohn designs and optimal optical orthogonal codes. For \( 3 \leq m \leq 8 \), the spectrum for \( V(m,t) \) has been determined. In this article, we investigate the existence of \( V(m,t) \) with \( m = 9 \) and show that a \( V(9,t) \) always exists in \( GF(q) \) for any prime power \( q = 9t + 1 \) with the exception of \( q = 73 \) and one possible exception of \( q = 5^6 \).