What can one say about the matrices M with the following properties: 1) the rows and the columns are indexed by the elements of a finite field with an even number q of elements; 2) All the matrix entries are either 1 or -1; 3) the sum of the entries in each row and sum of the entries in each column is zero. 4) the additive group of the field in (1) acts regularly on the rows and the columns of the matrix M and the action preserves M. that is, the (i,j)- entry of M is equal (g(i),g(j))-th entry of M for each element of the field.

As comprehensive an answer or a reference is very much appreciated. -N.S.N.Sastry, IIT Dharwad (nnsastry@gmail.com)