We give a new construction for a known family of weighing matrices using the 2-adjugate method of Vartak and Patwardhan. We review the existence of W(n,k²), k = 1,.. ,12, giving new results for k = 8,...12

Sarvate, D G

Seberry, Jennifer

English

Research Online

University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences 1988 On weighing matrices with square weights D G. Sarvate Jennifer Seberry University of Wollongong, jennie@uow.edu.au Follow this and additional works at: https://ro.uow.edu.au/infopapers  Part of the Physical Sciences and Mathematics Commons Recommended Citation Sarvate, D G. and Seberry, Jennifer: On weighing matrices with square weights 1988. https://ro.uow.edu.au/infopapers/1034 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 On weighing matrices with square weights Abstract We give a new construction for a known family of weighing matrices using the 2-adjugate method of Vartak and Patwardhan. We review the existence of W(n,k²), k = 1,.. ,12, giving new results for k = 8,...12. Disciplines Physical Sciences and Mathematics Publication Details Sarvate, DG and Seberry, J, On weighing matrices with square weights, Optimization, Design of Experiments and Graph Theory, (M. N. Gopalan and G. A. Patwardhan, (Eds)), Indian Institute of Technology, 1988, Powai, Bombay, 71-76. This journal article is available at Research Online: https://ro.uow.edu.au/infopapers/1034 On Weighing Matrices with Square Weights D.G.Sarvate and Jennifer Seberry University of Sydney We give a new construction for a known family of weighing matrices using the 2-adjugate method of Vartak and Patwardhan. We review the existence of Wen, kl), k = 1, ... ,12, giving new results for k '" 8, ... 12. 1. Introduction A weighing matrix. W = W(n,k), is a square matrix of order n, with entries 0,+1, or -I, with the pro-perty that the scalar product of distinct rows is zero. Thus wwT -kJ - '" It has been conjectured that Conjecture 1. (Seberry). When n~(mod4) there exists a W(n,k)jor all k::;; n. We discuss the conjecture Conjecture 2. For given k there exists an integer no. so thatfor every n > no. there exists a W(n,k). It is known that THEOREM 1 (Geramita. Geramita and Seberry[1975176]). If a W(n.k) exists and n is odd, then k is a square and We know that (a) a W( m2 + m + I, m 2) exists whenever m is a prime power and only if m is the order of a projective plane; (b) given any integer k2 there exists an n dependent on e so that for every m > n a W (m,k 2) exists; (c) there exists a (0 W(m,4). where m > 3 (except m",5,9, which do not exist); (ii) W(m,9). where m > 21 (m = 11 does not exist and m = 17,21 are undecided; m = 15 exists, Gibbons and Mathon[1986] found a W(19,9»; (iii) W(m,16), where m > 35 (m = 17, 19 do not exist and m = 23, 25, 27, 29, 33, 35 are unde-cided; m = 31 exists); iv) W(m,25), where m > 81 (m = 27,29 do not exist-- there are many undecided cases); (v) W(m,36), where m > 137; (m = 37 •... ,41 do not exist) (vi) W(m,49), where m > 199; (m "" 49, ... ,58 do not exist). The proof of these results is in Geramita and Seberry [1979, p324-325], We present a construction for weighing matrices using the 2-adjugation method of Vartak: and Patwardhan and present results for W(n,64),W(n,81),W(n,I00),W(n,121),W(n,144). This allows us to say (vii) W(n,64) exist for n > 135 and W(2n,64) exist for 2n > 3t (Lemma 8); W(n,64) for n = 65 • ... ,71 do not exist; (viii) W(n.SI} for n > 377, W(4n,Sl) for n > 39, W(130,81) and W(91,81) exist (Section 1 (a»; W(n,8t) for n = 8t, ... ,89 do not exist; (ix)W(n,I00) exist for all n > 336 and W(2n,loo) exist for all n > 59. (Lemma 10); W(n,IOO) forn = 101, ... ,109 do not exist; (x) W(n,12l) exist for all n > 513, W(2n,12l) exist for 2n > 381. W(4n,12l) exist for 4n > 259, W(8n,12l) exist for 8n > 127. (Lemma 11); W(n,I21) do not exist forn = 121, ... ,131; (xi) W(n,l44) exists for all n > 649, W(2n,144) exists for all2n > 377; (Lemma 12); W(n,144) do not exist for n = 145, ... ,155. 2. Known results We llse the following results THEOREM 2. (Dandawate). If 4t is the order of an Hadamard matrix then there exists a W(2t(4t-l), 4t2). THEOREM 3. (Geramita and Seberry) Let r be any number of the form 21l • lOb'26c'5d '13" , and let n be any integer> 2"·lOb·26c·6d ·14· , a,h,c,d,e nonnegative integers. Then there exist (a) orthogonal designs of order 4n and types (1,I,r,r) and (l,4,r,r); (h) an orthogonal design of order 2n and type (r,r) or a W(2n,2r). COROLLARY 4. Let 2n he any integer> 119 then there exists a W(2n,100). We use the fact that if there is an orthogonal design of type (l,k) in order n there is an orthogonal design of type (l,l,k,k) in order2n. Hence a W(2n,2k+l) exists. By the results of Geramita and Seberry[1979, Chapter 8] and Seberry[1982] we have THEOREM 5. Every orthogonal design of type (l ,Je), k < n-1, exists in order n = 21 ,2£+1'3.21+1'5. 2H1 '7, 21+1 '9,21+4-15, 2H4 ·21, t a positive integer. Hence W(2n.81) and W(2m,12l) exist for those m,n provided n > 40, m > 60. Thus W(16a,81) existfor a > 5 and W(16b,12l) existfor all b > 7 ( since hy Theorem 1.149 of Geramita and Seberry an OD(1,30) exists in every order4n, n ~ 11 we have OD(l,J ,2,60,60) and hence W(16n.121)for n ~ 11). Also from Theorem 1.149 of Geramita and Seberry we have LEMMA 6. Orthogonal designs of type (1.40) exist in all orders 4n, n > 10. Thus W(8n,81) exist n > 10. Since there are Golay sequences of length 40 we have LEMMA 7. There is an orthogonal design of type (1.1,40,40) in all orders 4n. n ~ 40. Thus W(4n,81) exist for n~40. 3. Some W(n, k 2 ) 3.1. Case W(n,64) The following exist W(2n,64), n>31 (Corollary 4), W(120,64) (Dandawate), W(73,64) (Section 1, (i). Thus we have LEMMA 8. There exist W(n,64)for n;,. 135. 3.2. Case W(n,81) The following exist W(4n.81), W(91,81), W(llO,81), W(l69,81), Thus we have n > 39 (Lemma 7) (complement of PG(9,2» (from W(IO,9) and W(13.9» (from W(ll,9)). LEMMA 9. There exist W(n,BI)/orn;,. 337. 3.3. Case W(n,lOO) W(2n,lOO) exist for all n > 59. (Corollary 4), W(m,25) and W(n,4) give a W(mn,l00) so since W(m,25) exist for m E {26,28,31,32,36,40,42,44,48, and all even orders> 51} and W(n,4) exist for n E {4,6,7.8,10,11 •... } we have W(mn,l00) for mn e {104,112.217}. Thus we have LEMMA 10. There exist W(n,IOO)/or all n > 336. W(2n,JOO) exist/or 2n = 104,112 and a1l2n > 119. 3.4. Case W(n,121) W(133,121) exists «i), section 1). W(16r2,121) exist for all t > 2 since W(4t,1l) exists. W(16b,121) exists for all b > 7 (from Theorem 5). W(122,121) exislS since 121 is a prime power. We construct a W(132,121). Let A be a (I,ll) design in order 12, replace the first variable by J 11, the matrix of ones, and the second variable by B where B = (bij ) = ( XU-i) ), X the quadratic character with X{O)=O. Then the new matrix is the required W(132.121). Thus we have LEMMA 11. There exist W(n,12l)/or all n > 513. W(2nJ21J exist/or 2n > 381. W(4n,121) exist/or 4n > 259. W(16n,121) exist for 16n > 127. 3.5 Case W(n,l44) Since there are 4-complementary sequences of length 9 and weight 36 and Golay sequences of length 4, we have 4-complementary sequences of length 36 and weight 144. Thus we have W(4n,144) exist for all n > 35. Hence W(273,144) exists as it is W(13,9) x W{21.16). W(234,144) exists as it is W(18,16) xW(13.9). LEMMA 12. W(n,144) existsfor all n > 649. W(2n,144) exists for a1l2n > 377. 4. A new construction for weighing matrices using 2.adjugates Vartak and Patwardhan[19711 and Patwardhan and Vartak[1980] constructed the 2-adjugate mod 2 class of designs for unreduced BIBDs and symmetrical BlBDs with A. =1. Patwardhan, Dandawate and Vartak[1984] applied the technique of 2-adjugation to a (O,l,-l) matrix and obtained a class of generalized PBIBDs with two associate classes and with triangular association scheme with unequal block sizes. In this section we use the concept of2-adjugates to obtain a construction for weighing matrices. The 2-adjugate of a (0,1,-1) matrix of N of order (v x b) is defined to be the matrix whose elements are the determinants of all possible 2 x 2 submatrices of N, arranged in lexicographic order. The 2-adjugate N'" of a matrix N whose entries come from a group is defined to be the matrix whose entries are formed by taking formal detenninants of all possible 2 x 2 submatrices of N. Here by formal determinant we mean the following: If the submatrix is x,y z,w then the corresponding entry of N* is xw-yz. Notice that in the special case when N = GH{p,EA(P», the generalized Hadamard matrix of order p, p a prime power, the (i,j)rJ. entry of N is r(i-lJU-l) and hence by replacing the group elements by the corresponding matrix representation the 2-adjugate of any arbitrary two by two submatrix r(i-I)(.r-I) r(i-I)(r-I) rU-1X.r-I) rU-1)(r-l) with i < j is r(i-l)(.r-I)+U-l)(I-I) _ r(i-l)(r-l}+(i-lX.r-I). An arbitrary row ofH* obtained from the irJ. and jrJ. rows ofH is given by r(i-l)(I-I)+{,i-1X.r-l) _ r(i-l)(4-1)+U-IXt-I) ; t>-l .... ,p-l; s=t+l, ... ,p. We prove the following LEMMA 13. Let H- be the 2-adjugateformedfromH .. GH(p.EA(p)), the generalized Hadamard matrix of size p, p a priml! power, then the inner product of any distinct rows of I-I· obtained from the i llt and P" rows and k and lilt rows of II respectively is zero and the inner product of a row obtainedfrom the i '" and jilt rows of /I with itselfis p2/ • pl. Proof: The inner product of two rows of /I' is given by 1=(P-I) I"P L L [T(i-I)(I-l}l{i-l){.r-I).T(i-I){s-I)+(j-I)(I-I)] ,,'1 .r:{I+I) . [TP-(k-I)(I-I}-(r-I X,-I) • TP -(k-l)(s-Ij-(r-I)(I-l)] 1=(P-I) .r"P 19.0-1) I~P = L L r(i-k)(I-I)+U-r)(.r-I) + L L T(i-k)(s-I)+U-r)(I-l) '=1 I:{I+I) ,,,1 .r:{I+I) 1=(p_I) .r=p ".(p-I) .r"'P . L L y(i-rXs-l}l{i-k)(I-l). L L r(i-..)(I-l)+(j-J;)(s-I) 1=1 .r={I+l) 1=1 I:{I+I) Notice that the first and second terms give 191>-1) '''P s=(P-i) 19' L L r(i-k)(H). T(j-r)(I-I) + L L r(i-J;)(I-I). r(j-rX.r-l) 1=1 1=(1+1) 1=1 1={I+l) Which gives us 19' '''P L [T(i-k)(I-l) • L T(j-"Xs-l)] (s:;tt). /.<1 1=1 Now we add and subtract '=-L T(i-J;)(H}I{i-l')(t-l). 1=1 Similarly we evaluate third and founh terms and get '''P l=p l=p ''''P '''P 1=(1 L r(i-J;)(t-l) • L yU-r)(I-l) • L y(H)(i+j-k-r) • L T(i-r)(I-I) • L + L T(I-1Xi+j-k-l') • 1=1 1=1 1=1 1=1 1=1 1=1 Which is plI . pJ when i=k andj=r, and zero otherwise. THEOREM 14. Weighing matrices W(p2(p_l), p2) existfor all prime powers p. Proof: Fonn H* from H = GH(p,EA(p)) the generalized Hadamard matrix.Obtain G from H* by replacing the entries of H* by the corresponding (0,1,-1) matrices.Form a matrix S = I x Ip . Then is the required weighing matrix. EXAMPLE. LetH .. GH(3,EA(3» given by [: ; ~l Replace ooi by Ti and 1 by the identity matrix of order '3, where T is given by [0 1 0] o 0 1 1 0 0 NowformH*: [1-ro l-ro' ro'~l 1_002 1-00 ro-oo2 002_00 ro-oo2 oo2-m which gives G: 1 0 0 1 o - 1 0 1 1 0 10-1 0 o 1 1 0 0 1 1 0 o 1 1 0 0 1 0 1 0 1 1 0 1 0 0 1 o 1 0 1 1 0 0 1 1 0 1 0 1 - 0 1 0 Now formS: 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Now fonn W: [~ ~l Note that in this particular case we have W(18. 32). We also note that in the case of order 3 we have using [0 1 1 1 1 0 1 -GW(4.3.2)= 1 _ 0 1 1 1 - 0 that 0000 1 1 1 1 1 1 1 1 1 o 1 1 1 000 1 1 1 o 1 1 1 1 1 1 000 011 1 111000 101 101 1 0 1 1 - 0 1 1 - 0 1 G 1 - 0 1 1 1 - 0 1 1 - 0 1 1 - 0 is a W(13,9). References Prabhakar N. Dandawate. (1983), On Designs with Generalized Incidence Matrices, PhD. thesis, I.I.T., Bombay. A.V.Geramita, J.M.Geramita and J.Seberry Wa1lls,(1975176), Orthogonal designs, Linear and Multilinear Algebra, 3, 281-306. A.V.Geramita and Jennifer Seberry, (1979), Orthogonal Designs: Quadratic Forms and Hadamard Matrices, Marcel Dekker, New York-Basel. Peter Gibbons and Rudolf Mathon, (1986), Construction methods for Bhaskar Rao and related designs, J AustraIMath.Soc. Ser A. (to appear) G.A.Patwardhan and M.N.Vartak, (1980)" On the adjugate of a symmetrical balanced incomplete block design with A. = I, Lecture Notes in Mathematics, Vol 885 Combinatorics and Graph Theory, Springer-Verlag. Berlin, 133-155. G.A-Patwardhan. P .N.Dandawate and M.N.Vartak, (1984), On the adjugate of a (0,-1,1) incidence matrix. Indian J. Pure and Appl. Math .• 15(6), 589-596. Jennifer Seberry, (1982). The skew-weighing matrix conjecture. Uni. of Indore ResearchJ. Science, 7, 1-7. MN.Vartak: and G.A.Patwardhan,(1971), The 2-adjugate mod 2 class of designs. Journal o/Combinatorial Theory, 11, 11-26. 

On weighing matrices with square weights

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On weighing matrices with square weights

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