Existence of $V(9,t)$ Vectors

Abstract

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 $3\le m\le 8$, 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 $q=5^6$.