Classification of Symmetric Hadamard Matrices Up to Order 32

Abstract

In this paper, symmetric Hadamard matrices are classified up to Hadamard equivalence for all orders at most 32. In particular, an error in the previous classification of symmetric Hadamard matrices of order 28 is corrected. The number of distinct symmetric Hadamard matrices at each order is also enumerated. The classification and enumeration are carried out using a new algorithm that determines whether a given Hadamard matrix is equivalent to a symmetric one. The extension of the algorithm to weighing matrices is also described, and theoretical results concerning symmetric Hadamard and weighing matrices are established.