New skew-Hadamard matrices via computational algebra

In this paper we formalize three constructions for skew-Hadamard matrices from a Computational Algebra point of view. These constructions are the classical 4 Williamson array construction, an 8 Williamson array construction and a construction based on OD(16; 1, 1, 2, 2, 2, 2, 2, 2, 2), a 9-variable full orthogonal design of order 16. Using our Computational Algebra formalism in conjunction with supercomputing, we perform exhaustive and partial searches for all three constructions. The computational results of these searches indicate a fundamental difference among these constructions, namely that the 8 Williamson array construction and the full orthogonal design construction exhibit exponential growths in the number of solutions. Subsequently, we analyze the computational results of these searches to locate inequivalent skew-Hadamard matrices, using the profile criterion. We show how to use the doubling construction to construct inequivalent skew-Hadamard matrices of order 2n from skew-Hadamard matrices of order n. Combining our computational results and the doubling construction we establish constructively 30 new lower bounds for the numbers of inequivalent skew-Hadamard matrices of orders 60, 68, 72, 76, 80, 84, 88, 92, 100, 104, 108, 112, 116, 120, 136, 144, 152, 160, 168, 176, 184, 200, 208, 216, 224, 232, 240, 288, 352, 416. All 236 I.S. KOTSIREAS AND C. KOUKOUVINOS the inequivalent skew-Hadamard matrices constructed in this paper are available on the web page http://www.cargo.wlu.ca/skew-Hadamard .