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Circulant Weighing Matrices Circulant Weighing Matrices by Goldwyn Millar A Thesis submitted to the Faculty of Graduate Studies In Partial Fulfillment of the Requirements for the Degree of MASTER OF SCItrNCtr Department of Mathematics University of Manitoba Winnipeg, Manitoba Copyright @ 2009 by Goldwyn Millar THE UNIVBRSTTY OF MANTTOBA FACULTY OF GRADUATB STUDIBS COPYRIGHT PBRMISSION Circulant Weighing Matrices B--v Goldwyn Millar A Thesis/Practicum submitted to thc Faculty of Graduate Studies of The Univer.sity of Mnnitob¿r in ¡rartitl fulfillure nt of thc requirernent of the degree of Master of Science Goldrvvn MillarO2009 Permission h¿rs been grirntetl to the Univcl'sit],of M¿rnitoba Libraries to le¡rd a copy of this thesis/¡rracticttm, to Librar¡' antl Arctrives C¿rn¿rtla (LAC) to lencl a copy of this thesiii¡rracticum, and to LAC's âgent (UMIiProQuest) to microfilm, sellcopies and to pi t lirtr an abstr.act of this thesis/prncticum. This reproductiotl or copy of this tllesis h¿rs been rn:rde available by authority of the copyright owner solely for the pur¡rose of ¡rrivate study and reseârch, *nd may onl¡, 1¡" reprottucert aria copiea as perrnitted b¡' co¡rvright larvs or rvith ext)rcss n'l'itten ¿ruthoriz¿rtion frour thà copyrig¡t on,nér. Abstract A ci,rculant matrin is a matrix such that each of its rows, after the first, can be obtained from the row above it by a right cyclic shift. A ci,rculant wei,gh'ing matri,r of order n and weight w (a CW(n,w)) is a n x n circuiant (0, t1)-matrix trZ such that WWr : wI. Circulant weighing matrices can be constructed by exploiting properties of quadrics in projective and affine space. Group characters provide a link between CW(n,w)'s and elements of absolute value 1/w rn Zle"l (the cyclotomic field over the nth roots of unity). By making use of this link, it is possible to translate facts about the decomposition groups of prime ideals in Z[("] into necessary conditions for the existence of CW(n,T.u)'s; this technique is the basis of several powerfui methods for studying CW(n,tu)'s: the Self-conjugacy method, the Multiplier method, and the Field descent method. I would like to thank my supervisor, Dr. Craigen, for 3 and tf2yearc of valu- able guidance and criticism and for the financial support he has provided for me throughout this entire process. I wouid like to extend my thanks to my thesis com- mittee, Dr. Davidson, Dr. Gunderson, and Dr. Li; for their valuable feedback. F\rrther, I am grateful to the University of Manitoba mathematics department for the partial financial support I received during the first two years of my studies. I would also like to thank my parents, Barbara and Scott Millar, and my partner, Molly Davidson, for all of their support and encouragement. Contents trntroduction 1 1.1 H¿r<lanl¿rrcl \{atrices 4 1.2 Ho1;c:lling's \\¡eighing Problern I 1.3 Sec¡urc:crs \\¡ith Good ;\utocorrelat,ion Properties 13 1.1 \4¡eighing N.fatrices of Oclcl Orcler t7 1.5 Group Deveiopercl \\,eighing lv'I¿rtricers 26 1.6 Ror,r'sulrrs 32 1..7 Negacvclir: \\¡eighing l,fatrir:es óó LB Ecluivalence ó ( 1.9 Eigenr,alues c.,f Grou¡t Dovclopecl ld¿rtriccs 39 1.10 Lalge Weiglrt Circiilant \\¡eighing l\4atrices 40 1.I1 Hankel !\¡eigliing N,lntrices 47 Constructíng Circulant Weighirrg Matrices 49 2.1 Prerecluisites . 51 2.1.7 F3acftgrourrcl fronr Projective Geometlv 51 '2.\.2 Backgrouncl From Affine Geometr.l, 55 2.7.3 Tho Tr¿rrxl Rrnction 58 2.2 Coristluctiug Cirr:ulzrnt \fu'eighing N,Izrtrices Lsing Projective Geornetr¡' S) 2.2.1 Cvclic Diflêrence Sets . 63 '2.2.2 Singerr's Thc<>rern 67 2.2.3 Quaclrics ancl Quaclr¿rtic; Sets 77 2.2.4 Buiklirrg Circul¿nt Wcighing \,'f¿rtricxrs lising Quiidrics 77 l.;t The Iì.ela.tive Difl'ert¡ncer Set Ccinstluctiorl 80 2.3.L Relative l)iffêr'ence Sets ancl Their Corurection to Group I)e- veioped \\ieighing trdatricers 80 2.3.2 Backglound fi'oln Characl;er Theoly 87 ¡l.fD .L.ù.Ð Thc W¿l,1sir-Ilacl¿nr¡,rrcl Tr ¿msfìrrrr 92 ¿.J.-r llre¡ Construct,ion fu q:2t 94 () .-) l: L.¿.,) Constr u cl. i ng N egacvcli c \,\'eigiri ng \,'1 a.t rices 102 Á.lgebraic Non-Existence Results LOT 3.1. Ba,ckgI'ouricl FÌ'onl Algebra,ic; Nurnber T'hec¡rr' 108 :1.2 lllhe Self-CJonjuga,c:y Met.hod L74 3.2.1 l\"I¿r's Lernm¿r 115 3.2.2 Applic;ations of the¡ Self'-Conjugtrc;¡r Ndethod . 120 il.3 1\.'Iultiplierrs ¿rncl the Classification of Cilcul¿rnt Weighing \datrices of Prirne Porver \Ä¡ciglrt L29 3.3.1. 'Jlype 1 l.'fultiplier Results 130 3.3.2 T¡'pe 2 il.'fultiplier Results 139 3.4 Thc Ficicl Descent \,'{et}rccl L52 3.4.1 Ïher \d¿rirr R.esult 153 3.4.2 A Iì.echictiorr Theolern 158 il1 :1..1.3 A IJouncl on Large \Ä¡eiglrt Circulant \1¡eighing N;l¿rtlir:es 161 A Original contributions L67 Bibliogrerplry L7O Index L82 tv Chapten 1 lntnoductioÍI Letw,n €Nand let w 1n. Ãwei,ghi,ng matri,r of. order n and wei,ght tl is a nxn (0, +1) matrix Iz7 such lhat WWr : wI . We may also refer to W as a W (n, w) or, alternatively, as a wei,ghi,ng matri,r ui,th paramaters (n,2,'). The row vectors of. W are mutually orthogonal and each contains exactly u) rroî-zero entries. The following matrix is aW(4,3) (throughout this thesis, -'s will be used to rep- resent -1's in matrices). 01 1- 10 11 1 01 10 Since every invertibie matrix commutues with its inverse, it follows tlnat W com- mutes withWr. Therefore, Wr is also a weighing matrix of weight u. So the column vectors of W are also mutually orthogonal and each contains exactly w rLorr-zero CHAPTER 1. INTRODUCTION entries. By taking the transpose of the above W(4,3), we obtain another W(4,3). 01 10 1- 11 01 1 10 If W0 can be obtained ftomW by a permutation of the rows and columns of L4l, then the rows of W6 are mutually orthogonal and they each contain exactly w rron-zero entries. Thus, Wo is aiso a weighing matrix of weight tu. Likewise,, if. W¡ can be obtained by multiplying some of the rows and columrs of.W by -1, then VZs is still a weighing matrix of weight æ. Two weighing matrices that can be obtained from one another by permutation of rows and columns or by multiplication of rows and columns by -1 are considered equ'iualent as wei,ghi,ng rnatri,ces. The following two matrices are equivalent 117(6,4)'s, since the second can be ob- tained from the first via a permutation of the first two rows and a multiplication of the second column by -1. 01 10 11 10 11 01 10 0-1 01 10 1 01 i0 10 1- 0- 10 110 1 0- 10 1 011 0- 10 11 0- 101 10 CHAPTER 1. INTRODUCTION A ci,rculant matri,r is a matrix such that each of its rows, after the first, is obtained from the row above it by a right cyclic shift. A circulant matrix is de- termined by its first row; if. A : is a circulant, then A can be written as [ø¿¡................] circ(ø11, an,...,atn). Following convention, we will refer to a circulant weighing matrix of order n and weight w a.s a CW(n,ar). The second matrix in the above example is a CW(6,4). Most research about weighing matrices is concerned with determining the parame- ters n and t¿ suchthat there exists aW(n,tl). However, other questionsregarding structure and equivalence (in various senses of the word) are also considered. The study of circulant weighing matrices is interesting within the broader context of the study of weighing matrices. For instance, for each prime power g, there exists a CW(q2 * q-l1,92) (see Chapter 2 for two, out of many possible, justifications of this fact); these weighing matrices are, in a sense, optimal (see Theorem 1.4,.2 below). Further, Antweiler, Bomer, and Luke [,E]31,r]ii] discovered, via a computer search, aCW(33,25); this fiiled an open entry in Craigen's weighing matrix table [1"]r'r.;lrii]. Arasu and Torban [{l'{}r}] later provided a theoretical explanation of the existence of this matrix. Circulant weighing matrices are also interesting objects in their own right. Weighing matrices in general and circuLant weighing matrices in particular have applications in statistics and engineering (see [,r1,rr;'i{Í], [iì{,;il;1], IiiriiT]'], and It't.]u:ì], for instance). The study of circulant weighing matrices began in the late 1960's and the early 1970's with the discovery that they can be constructed using projective geometries (see [Oiirii;î] and Iiii.,is?1]). Infact, because of the group structure underlying these matrices, results and techniques from diverse branches of mathematics, such as num- CHAPTER 1. INTRODUCTIO¡ü ber theory, character theory, and combinatorics, are used to study them. Research on circulant weighing matrices appears in both the Mathematics and Engineering literature, and this research has been particularly intensive during the last 10 years. The primary purpose of this thesis is to elucidate the major results and techniques from the literature on circulant weighing matrices.
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