We employ theoretical and computational techniques to construct new weighing matrices constructed from two circulants. In particular, we construct W(148, 144), W(152, 144), W(156, 144) which are listed as open in the second edition of the Handbook of Combinatorial Designs. We also fill a missing entry in Strassler’s table with answer “YES”, by constructing a circulant weighing matrix of order 142 with weight 100

Arasu, K T

Kotsireas, Ilias S

Koukouvinos, Christos

Seberry, Jennifer

English

Research Online

University of Wollongong Research Online Faculty of Informatics - Papers (Archive) Faculty of Engineering and Information Sciences January 2010 On circulant and two-circulant weighing matrices K T. Arasu Wright State University, Dayton, Ohio 45435 USA Ilias S. Kotsireas Wilfrid Laurier University, ikotsire@wlu.ca Christos Koukouvinos 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 Arasu, K T.; Kotsireas, Ilias S.; Koukouvinos, Christos; and Seberry, Jennifer: On circulant and two-circulant weighing matrices 2010, 43-51. https://ro.uow.edu.au/infopapers/824 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 circulant and two-circulant weighing matrices Abstract We employ theoretical and computational techniques to construct new weighing matrices constructed from two circulants. In particular, we construct W(148, 144), W(152, 144), W(156, 144) which are listed as open in the second edition of the Handbook of Combinatorial Designs. We also fill a missing entry in Strassler’s table with answer “YES”, by constructing a circulant weighing matrix of order 142 with weight 100. Keywords two, weighing, circulant, matrices Disciplines Physical Sciences and Mathematics Publication Details Arasu, K. T., Kotsireas, I. S., Koukouvinos, C. & Seberry, J. (2010). On circulant and two-circulant weighing matrices. Australasian Journal of Combinatorics, 48 43-51. This journal article is available at Research Online: https://ro.uow.edu.au/infopapers/824 On Circulant and Two-Circulant Weighing MatricesK. T. Arasu ∗† Ilias S. Kotsireas‡Department of Mathematics and Statistics Department of Physics and Computer ScienceWright State University Wilfrid Laurier UniversityDayton, Ohio–45435 Waterloo, Ontario N2L 3C5USA CanadaChristos Koukouvinos Jennifer Seberry §Department of Mathematics Centre for Computer and Information Security ResearchNational Technical University of Athens University of WollongongZografou 15773, Athens NSW 2522Greece AustraliaDecember 6, 2009AbstractWe employ theoretical and computational techniques to construct new weighing matri-ces constructed from two circulants. In particular, we construct W (148, 144), W (152, 144),W (156, 144) which are listed as open in the second edition of the Handbook of Combi-natorial Designs. We also fill a missing entry in Strassler’s table with answer ”YES”, byconstructing a circulant weighing matrix of order 142 with weight 100.1 IntroductionA weighing matrix W = W (n, k) of order n and weight k is a square matrix of order n withentries from {0, − 1, + 1} such thatWW T = k · Inwhere In is the n× n identity matrix and W T is the transpose of W .A circulant weighing matrix, W = CW (n, k), is a weighing matrix of order n and weight kin which each row (except the first row) is obtained from its preceding row by a right cyclicshift. We label the columns of W by a cyclic group G of order n, say generated by g.∗Research partially supported by grants from NSF and AFOSR.†The author thanks the Centre for Computer and Information Security Research, Univ of Wollongong forits hospitality during the time of this sabbatical research.‡Supported by grants from NSERC and SHARCnet§Supported by an ARC grant.1For any circulant weighing matrix W = CW (n, k) defineA = { gi | W (1, gi) = 1 , i = 0, 1, . . . , n− 1}and B = { gi | W (1, gi) = − 1 , i = 0, 1, . . . , n− 1} (1)It is easy to see that | A | + | B |= k.For a circulant weighing matrix, W = CW (n, k) it is well known that k must be a perfectsquare, (see [7], for instance), write k = s2 for some integer s.For more on weighing designs, weighing matrices and related topics refer to [5].It is known [5, 8] that:Theorem 1 A CW (n, k) can only exist if (i) k = s2, (ii) | A |= s2+s2 and | B |= s2−s2 , (iii)(n − k)2 − (n − k) ≥ n − 1 and (iv) if (n − k)2 − (n − k) = n − 1 then M = J −W ∗W isthe incidence matrix of a finite projective plane, (here J is the n× n matrix of all 1’s and ∗denotes the Kronecker product).For a multiplicatively written group G, we let ZG denote the group ring of G over Z. We willconsider only abelian (in fact, only cyclic) groups. For S ⊆ G, we let S denote the element∑x∈S x of ZG. For A =∑g agg and t ∈ Z, we define A(t) =∑g aggt.It is easy to see (see [1], [2] or [3] for details):Theorem 2 A CW = W (n, s2) exists if and only if there exist disjoint subsets A and B ofZn satisfying(A − B)(A − B)(−1) = s2. (2)We shall identify a W = CW (n, k) with its first row of the group ring element∑i W (1, gi)giin ZG.Definition 1 The support of a circulant matrix C of order n is defined as the setsupport C = {i |C(1, i) 6= 0, 1 ≤ i ≤ n}In this paper we use the following notations:1. a W (n, k) denotes a weighing matrix of order n and weight k;2. a CW (n, k) denotes a circulant weighing matrix of order n and weight k;3. DC(n, k) denotes two {0,±1} sequences of order n each and (total) weight k, that havePAF zero; (see [7] for the definition of PAF)4. a 2 − CW (2n, k) denotes a W (2n, k) constructed from two circulants whose first rowsare given by DC(n, k).22 New ResultsWe obtain an extension of the following theorem of Arasu and Dillon [1].Theorem 3 If there exists a CW (n, k) with n odd, then there exists a CW (2tn, 4k) for eachpositive integer t > 1.An extension of Theorem 3 is Theorem 2.3 in Arasu, Leung, Ma, Nabavi, Ray-Chaudhuri [2]Theorem 4 Let G be a group such that the center of G contains an element α of order2. Let B be a W (G, k) and let C ∈ Z[G] such that C has coefficients 0,±1 and η(C) is aW (G/ < α >, k) where η : G −→ G/ < α > is the natural epimorphism. If B, αB, C, αCare pairwise disjoint, thenA = (1− α)B + (1 + α)C (3)is a W (G, 4k).Remark The notation W (G, k) used in theorem 4 above refers to a weighing matrix that isdeveloped using the group G; we avoid giving its definition for the sake of brevity and referthe interested reader to [2] for further details. We only wish to stress that if G is a cyclicgroup, then the W (G, k) is indeed a CW (n, k) where n is the order of G.For convenience we provide an extension of Theorem 3 to cover the case t = 1; although amore general version is contained in Theorem 4.Definition 2 Two circulant matrices A and B of the same order are said to have disjointsupport, if (support A) ∩ (support B) = ∅.Theorem 5 Let n be an odd positive integer. If there exist two CW (n, k) with disjointsupports then there exists a CW (2n, 4k).Proof. Let A and B be two CW (n, k) with (support A) ∩ (support B) = ∅. Then AA(−1) =BB(−1) = k in Z[G], where G is “the” unique multiplicatively written group of order n. Let< t >= Z2 where t2 = 1. Then H = G× < t > is a cyclic group of order 2n.We defineW = (1 + t)A + (1− t)B.ThenWW (−1) = 2(1 + t)AA(−1) + 2(1− t)BB(−1) = 2(1 + t)k + 2(1− t)k = 4k.Since A and B have disjoint supports with coefficients 0,±1, it follows that W has coefficients0,±1. Hence, W defines the required CW (2n, 4k). 2Definition 3 Two matrices A and B of the same order are said to have disjoint support, ifA?B = 0, where ? denotes the Hadamard product (element-wise product) of the two matrices.3The above definition of disjoint support for arbitrary matrices (i.e. not necessarily circulant)boils down to the definition 2 of disjoint support for circulant matrices.Theorem 6 If A and B are two W (n, k) with disjoint support then, since AAT = BBT = kI[A + B A−BA−B A + B]is a W (2n, 4k).Note that theorem 6 is important since it does not require any structural assumptions (likecirculant on A or B) - any random weighing matrices with disjoint support will work.2.1 ApplicationsLet G =< x > where x71 = 1. ThenA(x) = x7 + x35 + x33 + x23 + x44 + x9 + x45 + x12 + x60 + x16 + x22 + x39 + x53 + x52 + x47−x− x5 − x25 − x54 − x57 − x6 − x30 − x8 − x40 − x58andB(x) = x11 + x55 + x62 + x26 + x59 + x18 + x19 + x24 + x49 + x32 + x27 + x64 + x36 + x38 + x48−x13 − x65 − x41 − x63 − x31 − x14 − x70 − x66 − x46 − x17define two CW (71, 25) with disjoint supports. Following the construction of Theorem 5, wedefine W = (1 + x71)A(x2) + (1− x71)B(x2) where we reduce modulo 2 · 71 the exponents ofthe polynomial W . Therefore, according to Theorem 5, W defines a CW (142, 100). In orderto provide an independent verification of this result, we give explicitly the first row of thisCW (142, 100) constructed using Theorem 5:- - 0 0 - 0 + 0 - - + - 0 + 0 - + + + 0 + + + + - - - - 0 0 0 + + - ++ - + - 0 0 0 - + - + - + + - 0 + - + + 0 - 0 + - + - 0 + 0 + 0 0 + +0 + - 0 0 + 0 + 0 - - - - 0 + 0 - + + + 0 - - + + + + + + 0 0 0 + + ++ - - - + 0 0 0 - + - + + - + - 0 - + - - 0 + 0 - - - + 0 - 0 + 0 0 -+ 0Remark 1 The existence of a CW (142, 100) was previously open, see Strassler [10].Remark 2 The first example of a CW (71, 25) was given by Strassler [9].3 Two-Circulants or Double Circulant ConstructionsWe now extend the ideas of Section 2 to the “two-circulants” case.4Definition 4 Two elements A and B of the group ring ZG, where G is a cyclic group of ordern, are said to define two-circulants, or double-circulants, of order n with weight k, writtenDC(n, k), if (i) the coefficients of A and B are in {0, 1,−1} and (ii) AA(−1) + BB(−1) = k.The following theorem is taken from [7].Theorem 7 Let A and B define a DC(n, k). Let circ(A) and circ(B) be the circulantmatrices whose first rows are A and B respectively. Then[circ(A) circ(B)circ(B)T −circ(A)T]gives a2− CW (2n, k) = W (2n, k).For a double circulant weighing matrix, 2−CW (2n, k) it is well known that k must be a sumof two squares.Theorem 8 Let G be a cyclic group of order n. Let A and B be DC(n, k).Suppose that A and B have “disjoint” supports and |G| is odd. Let < t >= Z2 where t2 = 1.Define H = G× < t > andC = (1 + t)A + (1− t)B and D = (1− t)A + (1 + t)B.Then C and D define a DC(2n, 4k).Proof. Note the coefficients of C and D are 0,±1. NowCC(−1) = 2(1 + t)AA(−1) + 2(1− t)BB(−1) and DD(−1) = 2(1− t)AA(−1) + 2(1 + t)BB(−1).Hence CC(−1) + DD(−1) = 4(AA(−1) + BB(−1)) = 4k, as desired. 23.1 ApplicationsWe now apply theorem 8 to construct three new double circulant weighing matrices DC(74, 144),DC(76, 144), DC(78, 144). We note that the existence of the corresponding W (148, 144),W (152, 144) was previously open, see Craigen’s table [4]. We also note that there exist sym-metric and skew-symmetric W (156, 144). We are also grateful to R. Craigen for pointing outthat W (156, 144) can be constructed by the method of weaving. However the existence of aDC(78, 144), hence a W (156, 144) constructed from two circulants, was open.Proposition 1 There exists a1. DC(37, 36) hence a DC(74, 144) and hence a W (148, 144);2. DC(38, 36) hence a DC(76, 144) and hence a W (152, 144);3. DC(39, 36) hence a DC(78, 144) and hence a W (156, 144);4. DC(19, 18) hence a DC(38, 72) and hence a W (76, 72);5. DC(31, 18) hence a DC(62, 72) and hence a W (124, 72).5Proof.1. Consider the following DC(37, 36) taken from [7]:A = + + - - 0 - 0 - + + 0 + 0 0 + + 0 + 0 + 0 0 - + 0 + 0 0 0 - 0 + 0 0 0 0 0B = 0 0 0 0 - 0 + 0 0 0 + 0 - - 0 0 - 0 - 0 + - 0 0 + 0 + + - 0 - 0 + + - + 0Since A and B have disjoint supports, C and D as defined in theorem 8 define aDC(74, 144). Now we apply theorem 7 to this double-circulant pair (C,D), therebyobtaining a weighing matrix of order 148 and weight 144 from two-circulants.2. Consider the following DC(38, 36) with disjoint support, computed via string sorting[6]A = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - 0 + 0 - - + - + - - 0 - + + + + + - 0 + 0 -B = + - + - - - + 0 - - + - - - - 0 + 0 + 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - 0 - 0Since A and B have disjoint supports, C and D as defined in Theorem 8 define aDC(76, 144). Now we apply theorem 7 to this double-circulant pair (C,D), therebyobtaining a weighing matrix of order 152 and weight 144 from two-circulants.3. Consider the following DC(39, 36) with disjoint support, computed via string sorting[6]A = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - - + - + - - - - + 0 + + 0 0 + 0 - 0 + 0 - 0 + +B = - - 0 + + + - - + - - + - 0 0 0 0 0 0 0 0 0 0 0 0 0 0 + - 0 - 0 - 0 - 0 - 0 0Since A and B have disjoint supports, C and D as defined in Theorem 8 define aDC(78, 144). Now we apply theorem 7 to this double-circulant pair (C,D), therebyobtaining a weighing matrix of order 156 and weight 144 from two-circulants.Remark. We also note that there exist known but unpublished W (156, 144).4. Consider the following DC(19, 18) taken from [7]:A = 0 0 - 0 0 0 + + - 0 0 0 0 + + + 0 - +B = 0 0 - 0 0 0 - - - 0 0 0 0 + - + 0 - +If we reverse the second sequence we see that the resulting sequences have disjointsupports. The corresponding polynomials are:A(x) = x19 − x18 + x16 + x15 + x14 − x9 + x8 + x7 − x3,B(x) = −x17 − x13 − x12 − x11 + x6 − x5 + x4 − x2 + x.Following the construction of Theorem 8, we define C = (1+x19)A(x2)+(1−x19)B(x2),D = (1−x19)A(x2)+(1+x19)B(x2) where we reduce modulo 2 ·19 the exponents of thepolynomials C, D. Therefore, according to Theorem 8, C,D define a DC(38, 72), i.e.a 2−CW (76, 72) constructed from two circulants. In order to provide an independentverification of this result, we give explicitly the first rows of C,D (note that they haveidentical supports)60 + + - + - + + + - + + + + + + - - + 0 - - + - - - - + + + - + + - + - - +0 + - - - - - + - - - + - + - + + - - 0 + - - - + - + + - + + + - - - - + +5. Consider the following DC(31, 18)A = 0 0 0 0 0 0 0 - 0 - 0 0 0 0 0 - 0 + + 0 0 0 0 + 0 0 0 - 0 - -B = 0 - - + 0 0 0 0 - 0 0 0 - + 0 0 0 0 0 - 0 0 0 0 - + 0 0 0 0 0and use it as in 4. to obtain a DC(62, 36) and hence a 2− CW (124, 72)Note that the first rows of the circulant matrices C and D have identical supports. 2Remark. We note that circulant and double circulant weighing matrices have structure thatis amenable to Signal Processing [11] for wireless communications.4 AcknowledgmentsWe wish to thank R. Craigen for pointing out some important implications of the methodof weaving, for the construction of weighing matrices. We also wish to thank an anonymousreviewer for pointing out that circulant and two-circulant weighing matrices where n >> kare used in acoustic engineering.References[1] K. T. Arasu and J. F. Dillon Perfect ternary arrays. Difference sets, sequences and theircorrelation properties (Bad Windsheim, 1998), 1–15, NATO Adv. Sci. Inst. Ser. C Math.Phys. Sci., 542, Kluwer Acad. Publ., Dordrecht, 1999.[2] K. T. Arasu, K. H. Leung, S. L. Ma, A. Nabavi, D. K. Ray-Chaudhuri Circulant weighingmatrices of weight 22t, Des. Codes and Cryptography, (2006), 111–123.[3] K. T. Arasu, J. F. Dillon, D. Jungnickel and A. Pott, The solution of the Waterlooproblem, J. Comb. Th.(A), 17, (1995), 316–331.[4] R. Craigen, H. Kharaghani, Orthogonal designs, in Handbook of Combinatorial Designs.Edited by C. J. Colbourn and J. H. Dinitz. Second edition. Discrete Mathematics and itsApplications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2007. pp. 280–295.[5] A. V. Geramita and J. Seberry, Orthogonal designs. Quadratic forms and Hadamardmatrices, Lecture Notes in Pure and Applied Mathematics, 45, Marcel Dekker Inc. NewYork, 1979.[6] I. S. Kotsireas, C. Koukouvinos, J. Seberry, Weighing matrices and string sorting. Annalsof Combinatorics, 13, (2009), 305-313[7] C. Koukouvinos, J. Seberry. New weighing matrices and orthogonal designs constructedusing two sequences with zero autocorrelation function—a review. J. Statist. Plann.Inference 81 (1999), no. 1, 153–182.7[8] J. Seberry Wallis and A. L. Whiteman, Some results on weighing matrices, Bull. Austral.Math. Sec. 12, (1975), 433–447.[9] Y. Strassler, New circulant weighing matrices of prime order in CW (31, 16), CW (71, 25),CW (127, 64), paper presented at the R. C. Bose Memorial Conference on StatisticalDesign and Related Combinatorics, Colorado State University, 7-11 June, 1995.[10] Y. Strassler, The Classification of Circulant Weighing Matrices of Weight 9, PhD thesis,Bar-Ilan University, Ramat-Gan, Israel 1997.[11] L.C. Tran, T.A. Wysocki, A. Mertins, J. Seberry, Complex Orthogonal Space-TimeProcessing in Wireless Communications, Springer-Verlag, Norwell, USA, 2006.8

On circulant and two-circulant weighing matrices

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On circulant and two-circulant weighing matrices

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