Uniqueness of Butson Hadamard matrices of small degrees

Let BHn×n(m)BHn×n(m) be the set of n×nn×n Butson Hadamard matrices where all the entries are m  -th roots of unity. For H1,H2∈BHn×n(m)H1,H2∈BHn×n(m), we say that H1H1 is equivalent   to H2H2 if H1=PH2QH1=PH2Q for some monomial matrices P and Q whose nonzero entries are m  -th roots of unity. In the present paper we show by computer search that all the matrices in BH17×17(17)BH17×17(17) are equivalent to the Fourier matrix of degree 17. Furthermore we shall prove that, for a prime number p  , a matrix in BHp×p(p)BHp×p(p) which is not equivalent to the Fourier matrix of degree p gives rise to a non-Desarguesian projective plane of order p.