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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. For m=3,4,5,6 and 7 the spectrum for V(m,t) has been determined. In this article, we use Weil's theorem on character sums to show that a V(m,t) always exists in GF(mt+1) for any prime power q=mt+1 such that m and t are not both even and q>B(m)=[(E+E 2 +4F)/2] 2 , where...