Uniqueness of Butson Hadamard matrices of small degrees

Abstract

Let BHn×n(m) be the set of n×n Butson Hadamard matrices where all the entries are m-th roots of unity. For H1,H2∈BHn×n(m), we say that H1 is equivalent to H2 if H1=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) 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) which is not equivalent to the Fourier matrix of degree p gives rise to a non-Desarguesian projective plane of order p.