A construction for orthogonal designs with three variables

Transcription

1 University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 1987 A construction for orthogonal designs with three variables Jennifer Seberry University of Wollongong, jennie@uow.edu.au Publication Details Seberry, J, A construction for orthogonal designs with three variables, Combinatorial Design Theory, (C.J. Colbourne and R.A. Mathon, (Eds)), North Holland, 1987, Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library: research-pubs@uow.edu.au

2 A construction for orthogonal designs with three variables Abstract We show how orthogonal designs OD(48p²t;16p²t, 16p²t,16p²t) can be constructed from an Hadamard matrix of order 4p and an OD(4t;t,t,t,t). This allows us to assert that OD(48p²t; 16p²t,16p²t,16p²t) exist for all t,p 102 except possibly for tє{67,71,73,77,79,83,86,89,91,97}. These designs are new. Disciplines Physical Sciences and Mathematics Publication Details Seberry, J, A construction for orthogonal designs with three variables, Combinatorial Design Theory, (C.J. Colbourne and R.A. Mathon, (Eds)), North Holland, 1987, This journal article is available at Research Online:

3 Annals of Discrete Mathematics 34 (1987) Elsevier Science Publishers B.V. (North Holland) 437 A Construction for Orthogonal Designs with Three Variables Jennifer Seberry Department of Computer Science University College The University of New South Wales Australian Defence Forces Academy Canberra, A.C.T AUSTRALIA TO J.u:r ROSJ. 0)1 JUS l1truth BIR. TJlOJ.Y ABSTRACT We show how orthogonal designs OD(4Sp Z t;16p Z t, 16p 2 t,16p 2 t) can be constructed from an Hadamard matrix of order 4p and an OD~4tjt,t,t,t). This allows us to assert that OD{4Sp 2 tj 16p t,16p2t,16p2t) exist ror all t,p:::; 102 except possibly for te{67,71,73,77,7q,83,86,sq,q1,q7}. These designs are new. 1. Introduction Let H = (h jj ) be a matrix of order h with h ij E{l,-I}. H is called an Hadamard matrix of order n, if HHT '"" hlh where Ih denotes the identity matrix order of h. An orthogonal design A, of order H, type (P"P2"",p,..), denoted OD(njp"pz,...,Pu), on the commuting variables (±x,,±xv '",±Xu,O) is a square matrix of order 11 with entries ±xk where each Xk occur Pk times in each row and column such that the rows are pairwise orthogonal. In other words AA T,.,. (PIX? Pux~)I". It is known that the maximum number of variables is an orthogonal design is p(n), the' Radon number, where for n=z'b,b odd, set a=4c+d,o:::;d<4, then p(n)=sc +2d. OD(4tjt,t,t,t), otherwise called Baumert-Hall arrays, and OD(2 P ja,b,2 P -a-b) have been extensively used to construct Hadamard matrices and weighing matrices. For details see Geramita. and Seberry (HI7Q).

4 438 J. &berry Geramita, Geramita and Wallis (=Seberry) observed (see Geramlta and Sebcrry [HI79, 4.3]) that if A,B,C,D are four circulant or type 1 matrices of order n satisfying AAT + BBT + CC T + DDT = (~ Pix;) In i_i then A,B,C,D can be used in the Goethals-Seidel array or ( J. Seberry Wallis Whiteman array) A BR CR -DR -BR A DTR -etr CR _DTR A BTR (1). DR CTR _BTR A to form an orthogonal design OD(4niPt,pz,""Pu)' 2. Background Kharaghani (UI85) defined Ot '""!hij.hkj[ and applying that to Hadamard matrices of order 4p, obtruned matrices satisfying CjCj=O,i+i (2). :E Cl-(4p)214p' He then used this to show there are Bush-type (blocks J4p down the diagonal) and Szekeres-type (h ij = -1 =} h jj - 1 and not necessarily vice versa) Hadamard matrices. By using a symmetric Latin square he could also have shown that regular symmetric Hadamard matrices with constant diagonal of order (4p)2 could be constructed by his method. Hammer, Sal"Vate and Seberry applied Kharaghani's method to OD(n;sl"..,.'I ) and in particular OD(4t;t,t,t,t) and OD4s;s,s,s,s) obtaining existence of OD(48s~t t, t, t, 12s2t) and OD(808 2 t; 20s 2 t, 2Os 2 t, t, 2Os 2 t) from OD(4s;s,s,s,s) and OD(4t;t,t,t,t). Seberry (to appear) extended this further to obtain OD(16k8 2 t ;4ks 2 t,4k8 2 t,4ks 2 t,4k8 2 t) for {1,3,5,... }. We modify their techniques to obtain new orthogonal designs. 3. Construction Let G l, O 2,.,G4p be the Kharaghani matrices of order 4p obtained from an orthogonal design OD(4p;p,p,p,p). Let a,b,c be commuting variables and write [aol: ac2:,, :aop : bcp+l :... :bc2p : cc2p+l:.. :cc3p l [ac 3P +l:,,:ag 4p : ccp+ 1 :.. :cc2p : b02p+l:... :bg3p]

5 OrthogOMI De8igns with Three Variables.39 for the first rows of two block circulant matrices WI and W z. It can be checked that w 1 wf = wzwi. Write [ac2p+1: :acap: bc1: :bcp : ccap+1: :CC4"J [ac"+i:.. :ac2p : bci: bc,,: bc3p+ 1 : bc 4 "J for the first rows of two block bru:k-circulant matrices Wa and W 4 It can be checked that wawf = w 4 wf. Now by virtue of being circulant and bru:k-circulant Example. Let p = ie(i,2), je(3,.). 3 so there are 12 matrices of order 12, C1>...,C I2. Then WI = aclac2ac3bc4bcsbcscc7ccgccg acaaclaczbcsbc4bcsccgcc7ccg ac2acaaclbcsbcsbc4ccgcc9cc7 CC7cCgcCgaOlaC2aCabC4bCsbCs cog C07 cgg a03 aoi ao z b0 6 b0 4 bos cog cog c07 ao z ac a ao l bo s b0 6 b0 4 b04bosbcsc07cogcc9aolaczaoa bcsb04boscc9c07cosaoaaola02 bcsbosb04cggcg9cc7aozagaaol W 2 = aoio ag ll ac I2 cc 4 cos cc s b0 7 bog bog acl2 aoio aoll C06 cc 4 cos bog b07 bcg ac ll aol2 ac IO cc s cos cc 4 bog bog b0 7 bg 7 bgg bcg agio aoll aol2 C04 cos cos bg g bc 7 bcg ac I2 acio ag ll cc s cc 4 cgs bog bog bc7 ac ll aolz ac IO cos ccs c04 C04 ccs cos b0 7 bog bog aoio oc ll aol2 cos c0 4 cos bcg bc 7 bog agio ac ll ccs ccs cc4 bcs bcg bc7 aoll aol2 acio wlwf=iq Xbc t C;eT ;_4 but OJ Or is symmetric and so WI wf = W2 wi. Wa and W 4 are formed similarly but are back-circulant in blocks (i.e., type 2) in the language of Seberry-Wallis and Whiteman). Thus WI' W 2, Wa, W( are VVilJiamson-type matrices of order 12p2 they can be used to replace the variables of an OD(4i1,1,1,1) to get an OD(48p2; 16p2, 16p2, 16p2, 16p2). In general they can be used to replace the variables of an OD(4tit,t,t,t) so we have

6 440 J. Seberry Theorem: If there is an Hadamard matrix of order 4p and an OD(4tjt,t,t,t) then there is an OD(48p2tj16p2t,16p2t,16p2t,16p2t). Since Hadamard matrices of order 4p exist for all p ::s: 102 and OD(4tjt,t,t,t) exist for all t $102 except possibly for tes, S={67,71,73,77,79,83,86,S9,91,97} (see Sebefry (HIS6a)) we have Corollary; OD{48p2tj16p2t,16p2t,16p2t,16,.h) exist for all t,p $102 except pobsibly fot t ES. Acknowledgement The author's research is supported in part by grants from ACRB and ATERB. References [1] J. Cooper and J. Seberry Wallis, "A construction of Hadamard arrays", Bull. Austral Math. Soc., 7 (1972) [2J A.V. Geramita and J. Seberry, Orthogonal Designs: Quadratic Forms and Hadamard Matrices Marcel Dekker, New York-Basel (198B). [3] J. Hammer, D.G. Sarvate, J. Seberry, "A note on orthogonal designs", Ars Combinatoria, submitted. [4J H. Kharagani, "New class of weighing matrices", Ars Combinatorica, Vol. 19, pp [5J J. Seberry, "More towards proving the Hadamard conjecture", (preprint). [6] J. Seberry-Wallis, "Hadamard matrices, Part IV of W.D. Wallis, A. P. Street, and J. Seberry-Wallis", Combinatorics: IWorn Square, sum free sets and Hadamard matriees, Lecture Notes in Mathematics, Vol. 292, Springer-Verlag, Berlin Heidelberg New York.