Perfect binary arrays and difference sets

AbstractA perfect binary array is an r-dimensional array with elements ± 1 such that all out-of-phase periodic autocorrelation coefficients are zero. Such an array is equivalent to a Menon difference set in an abelian group. We give recursive constructions for four infinite families of two-dimensional perfect binary arrays, using only elementary methods. Brief outlines of the proofs were previously given by three of the authors. Although perfect binary arrays of the same sizes as two of the families were constructed earlier by Davis, the sizes of the other two families are new