Abstract
Skew-Hadamard matrices are of special interest due to their use, among others, in constructing orthogonal designs. In this paper, we give a survey on the existence and equivalence of skew-Hadamard matrices. In addition, we present some new skew-Hadamard matrices of order 52 and improve the known lower bound on the number of the skew-Hadamard matrices of this order.
References (41)
- A.H. Baartmans, C. Lin, W.D. Wallis, Symmetric and skew equivalence of Hadamard matrices of order 28, Ars. Combin. 41 (1995) 25-31.
- D. Blatt, G. Szekeres, A skew Hadamard matrix of order 52, Canad. J. Math. 21 (1969) 1319-1322.
- J. Cooper, J. Milas, W.D. Wallis, Hadamard equivalence, Combinatorial Mathematics, Lecture Notes in Mathematics, vol. 686, Springer, Berlin, Heidelbert, New York, 1978, pp. 126-135.
- R. Craigen, Hadamard matrices and designs, The CRC Handbook of Combinatorial Designs, in: C.J. Colbourn, J.H. Dinitz (Eds.), CRC Press, Boca Raton, Fl, (1996), 370-377.
- D.Z. Djokovic, Skew Hadamard matrices of order 4 × 37 and 4 × 43, J. Combin. Theory Ser. A 61 (1992) 319-321.
- D.Z. Djokovic, Ten new orders for Hadamard matrices of skew type, Univ. Beograd. Pupl. Electrotehn. Fak. Ser. Math. 3 (1992) 47-59.
- D.Z. Djokovic, Construction of some new Hadamard matrices, Bull. Austral. Math. Soc. 45 (1992) 327-332.
- D.Z. Djokovic, Good matrices of order 33, 35 and 127 exist, J. Combin. Math. Combin. Comput. 14 (1993) 145-152.
- D.Z. Djokovic, Five new orders for Hadamard matrices of skew type, Australasian J. Combin. 10 (1994) 259-264.
- D.Z. Djokovic, Six new orders for G-matrices and some new orthogonal designs, J. Combin. Inform. System Sci. 20 (1995) 1-7.
- D.Z. Djokovic, Skew-Hadamard matrices of order 188 and 388 exist, submitted.
- C. Elster, A. Neumaier, Screening by conference designs, Biometrika 82 (1995) 589-602.
- R.J. Fletcher, C. Koukouvinos, J. Seberry, New skew-Hadamard matrices of order 4 • 49 and new D-optimal designs of order 2 • 59, Discrete Math. 286 (2004) 251-253.
- S. Georgiou, C. Koukouvinos, On circulant G-matrices, J. Combin. Math. Combin. Comput. 40 (2002) 205-225.
- S. Georgiou, C. Koukouvinos, Some results on orthogonal designs and Hadamard matrices, Int. J. Appl. Math. 17 (2005) 433-443.
- S. Georgiou, C. Koukouvinos, J. Seberry, On circulant best matrices and their applications, Linear and Multilinear Algebra 48 (2001) 263-274.
- S. Georgiou, C. Koukouvinos, S. Stylianou, On good matrices, skew Hadamard matrices and optimal designs, Comput. Statist. Data Anal. 41 (2002) 171-184.
- S. Georgiou, C. Koukouvinos, S. Stylianou, New skew Hadamard matrices and their application in edge designs, Utilitas Math. 66 (2004) 121-136.
- S. Georgiou, C. Koukouvinos, S. Stylianou, Construction of new skew Hadamard matrices and their use in screening experiments, Comput. Statist. Data Anal. 45 (2004) 423-429.
- A.V. Geramita, J. Seberry, Orthogonal Designs: Quadratic Forms and Hadamard Matrices, Marcel Dekker, New York, Basel, 1979.
- J.M. Goethals, J.J. Seidel, A skew Hadamard matrix of order 36, J. Austral. Math. Soc. 11 (1970) 343-344.
- H. Kharaghani, B. Tayfeh-Rezaie, A Hadamard matrix of order 428, J. Combin. Designs 13 (2005) 435-440.
- C. Koukouvinos, J. Seberry, On G-matrices, Bull. ICA 9 (1993) 40-44.
- S. Kounias, T. Chadjipantelis, Some D-optimal weighing designs for n ≡ 3 (mod 4), J. Statist. Plann. Inference 8 (1983) 117-127.
- R.E.A.C. Paley, On orthogonal matrices, J. Math. Phys. 12 (1933) 311-320.
- J. Seberry Wallis, (v, k, ) configurations and Hadamard matrices, J. Austral. Math. Soc. 11 (1970) 297-309.
- J. Seberry Wallis, A skew-Hadamard matrix of order 92, Bull. Austral. Math. Soc. 5 (1971) 203-204.
- J. Seberry Wallis, Some (1, -1) matrices, J. Combin. Theory Ser. B 10 (1971) 1-11.
- J. Seberry Wallis, On supplementary difference sets, Aequationes Math. 8 (1972) 242-257.
- J. Seberry Wallis, On Hadamard matrices, J. Combin. Theory Ser. A 18 (1975) 149-164.
- J. Seberry Wallis, Construction of Williamson type matrices, Linear and Multilinear Algebra 3 (1975) 197-207.
- J. Seberry Wallis, On skew Hadamard matrices, Ars Combin. 6 (1978) 255-275.
- J. Seberry Wallis, A.L. Whiteman, Some classes of Hadamard matrices with constant diagonal, Bull. Austral. Math. Soc. 7 (1972) 233-249.
- J. Seberry, M.Yamada, Hadamard matrices, sequences and block designs, in: J.H. Dinitz, D.R. Stinson (Eds.), Contemporary Design Theory-A Collection of Surveys, Wiley, New York, 1992, pp. 431-560.
- E. Spence, Skew-Hadamard matrices of order 2(q + 1), Discrete Math. 18 (1977) 79-85.
- G. Szekeres, Cyclotomy and complementary difference sets, Acta Arith. 18 (1971) 349-353.
- G. Szekeres, A note on skew type orthogonal ±1 matrices, in: A. Hajnal, L. Lovász, V.T. Sòs (Eds.), Combinatorics, Colloquia Mathematica Societatis, Janos Bolyai, vol. 52, North-Holland, Amserdam, 1988, pp. 489-498.
- W.D. Wallis, A.P. Street, J. Seberry Wallis, Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices, Lecture Notes in Mathematicsa, vol. 292, Springer, Berlin, Heidelberg, New York, 1972.
- A.L. Whiteman, An infinite family of skew-Hadamard matrices, Pacific J. Math. 38 (1971) 817-822.
- A.L. Whiteman, Skew Hadamard matrices of Goethals-Seidel type, Discrete Math. 2 (1972) 397-405.
- X.-M. Zhang, G-matrices of order 19, Bull. ICA 4 (1992) 95-98.