Hadamard matrices of order 764 exist

Two Hadamard matrices of order 764 of Goethals– Seidel type are constructed. Recall that a Hadamard matrix of order m is a {±1}-matrix A of size m × m such that AA T = mI m , where T denotes the transpose and I m the identity matrix. We refer the reader to one of [2, 4] for the survey of known results about Hadamard matrices. In our previous note [1], written about 13 years ago, we listed 17 integers n ≤ 500 for which no Hadamard matrix of order 4n was known at that time. Two of these integers were removed in that note and the smallest one, n = 107, was removed recently by Kharaghani and Tayfeh-Rezaie [3]. Among the remaining 14 integers n only four are less than 1000. The problem of existence of Hadamard matrices of these four orders, namely 668, 716, 764 and 892, has been singled out as Research Problem 7 in the recent book [2] by Kathy Horadam. In this note we shall remove the integer 764 from the mentioned list by constructing two examples of Hadamard matrices of Goethals–Seidel type of that order. (We have constructed a bunch of examples but we shall present only two of them.) Consequently, the revised list now