Abstract
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 . For m =3,4,5 and 6 the spectrum for V ( m , t ) has been determined recently by Ling et al. In this article, Weil's theorem on character sums is used to get the spectra for V ( m , t )'s, for m =7. For variant V ( m , t )'s, such as V (2) ( m , t ) with m =2,4,6 and V (4) ( m , t ) with m =2,4, the spectrums are also determined. Three infinite families of V λ ( m , t )'s with λ=2, m=2; λ=2, m=3 and λ=3, m=2 are proved to exist.
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