Annals of Combinatorics 1 (1997) 377-389Properties of Complete Sets of Mutually Equiorthogonal Frequency Hypercubes Ilene H. Morgan Department of Mathematics and Statistics, University of Missouri-Rolla
Rolla, MO 65409-0020, USA
Received July 15, 1997 AMS Subject Classification: 05B30 Abstract. A complete set of mutually equiorthogonal frequency hypercubes (MEFH) of order n and dimension d, using m distinct symbols, has (n-1)^{d}/(m-1) hypercubes. In this article, we explore the properties of complete sets of MEFH. As a consequence of these properties, we show that existence of such a set implies that the number of symbols m is a prime power. We also establish an equivalence between existence of a complete set of MEFH and existence of a certain complete set of Latin hypercubes and a certain complete orthogonal array. Keywords: frequency hypercubes, frequency squares, Latin hypercubes, Latin squares, orthogonal arrays, Hadamard matrices References 1. R.C. Bose and K.A. Bush, Orthogonal arrays of strength two and three, Annals Math. Stat. 23 (1952) 508–524. 2. J. Dènes and A.D. Keedwell, Latin Squares and Their Applications, Academic Press, New York, 1974. 3. J. Dènes and A.D. Keedwell, Latin Squares, Annals of Discrete Math., Vol. 46, North-Holland, Amsterdam, 1991. 4. W.T. Federer, On the existence and construction of a complete set of orthogonal F(4t;2t,2t)-squares design, Annals Stat. 5 (1977) 561–564. 5. A. Hedayat and W.D. Wallis, Hadamard matrices and their applications, Annals Stat. 6 (1978) 1184–1238. 6. P. Höhler, Eine Verallgemeinerung von Orthogonalen Lateinischen Quadraten auf Höhere Dimensionen, Diss. Dokt. Math. Eidgenöss, Hochschule Zürich, 1970. 7. C.F. Laywine, Complete sets of orthogonal frequency squares and affine resolvable designs, Utilitas Math. 43 (1993) 161–170. 8. J.P. Mandeli, F.-C.H. Lee, and W.T. Federer, On the construction of orthogonal F-squares of order n from an orthogonal array (n,k,s,2) and an OL(s,t) set, J. Stat. Plann. and Inf. 5 (1981) 267–272. 9. I.H. Morgan, Equiorthogonal frequency hypercubes: preliminary theory, Designs, Codes \& Cryptography 13 (1998) 177–185. 10. I.H. Morgan, Construction of complete sets of mutually equiorthogonal frequency hypercubes, Discrete Math., to appear. 11. G.L. Mullen, Polynomial representation of complete sets of frequency squares of prime power order, Discrete Math. 69 (1988) 79–84. 12. A.P. Street and D.J. Street, Combinatorics of Experimental Design, Oxford Sci. Publ., Clarendon Press, Oxford, 1987. |