Higher dimensional orthogonal designs and applications

When n2 elements are given, they can be arranged in the form of a square; similarly, when n8 elements (g ≥ 3 an integer) are given, they can be arranged in the form of a g-dimensional cube of side n (in short, a g-cube). The position of the elements can be indicated by g suffixes. Disciplines Physical Sciences and Mathematics Publication Details Hammer, J and Seberry, J, Higher dimensional orthogonal designs and applications, IEEE Transactions on Information Theory, Vol. IT-27(6), 1981, 772-779. This journal article is available at Research Online: http://ro.uow.edu.au/infopapers/1005 772 IEEE TRANSACTiONS ON INfORMATION TIiEOlty, VOL. IT-27, No.6, NOVEMBER 1981 Higher Dimensional Orthogonal Designs and Applications JOSEPH HAMMER AND JENNIFER R. SEBERRY [n memory of Professor G. Gaspar Abstract-11Kconcept of orthogonal design is extended to higher IUmensrons. A proper g-djmensionaJ design [d'jk oJ is defined as one in which all panIlel (g l):dimensionallayen, in any orientation parallel to a hype!" plane, are uncorrelated.. This is equivalent to the requirement that d,)k' v E {O, ±x1,"··, ±x.}, where XI'" ',x, are commuting variables, .......11Kconcept of orthogonal design is extended to higher IUmensrons. A proper g-djmensionaJ design [d'jk oJ is defined as one in which all panIlel (g l):dimensionallayen, in any orientation parallel to a hype!" plane, are uncorrelated.. This is equivalent to the requirement that d,)k' v E {O, ±x1,"··, ±x.}, where XI'" ',x, are commuting variables, ....... where (sl" . " $,) are integers giving tbe occurrences of ±x1,"', ±x, in each row and oolwnn (this is called the type (Sj,"',S,)K-l) and (pqr··· yz) rqwesents aU pemrutations of (ijk'" vJ. This extends an idea of Paul J. ShIiclrta, lI'hose higher dimensional Hadamard matrices are special cases with Xl"" x, E {I, -\}' ($1" ',s,) = (g), and (~,SIX:) = g. Another special case is higher dimensional weighing matrices of type (k )g, which have XI" .. , x, E {G, I, -\}, (s,,' . ',8,) = (k), and (I, SI-"}) = k. SbIichta found proper g-dimensiooal Hadamard matrices of size (2'Jg. Proper orthogonal designs of type (1,1)3 aDd (I, I, I, 1)3 art' used to obtain lUgbel' dimensional orthogonal designs. Hadamard matrices, and weighing matrices. A possible approach to using higher dimensional weighing matrices aDd Hadamard matrices in codes is discussed, as weD ru. their connection with higher dimensional orthogonal functions (Walsh, Haar, etc.),