[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-144477-105":59,"doc-detail-144477-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","orthogonal-designs-from-negacyclic-matrices","Orthogonal Designs from Negacyclic Matrices","","This paper studies how negacyclic matrices can be used to construct orthogonal designs, which in turn yield Hadamard matrices. Results are provided for all possible parameter tuples for orthogonal designs of order 12, with only a small number of exceptions for orders 20 and 28. The work develops algebraic structure using negashift and related matrices, and derives properties governing inner products between rows of negacyclic matrices.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/orthogonal-designs-from-negacyclic-matrices/144477/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/orthogonal-designs-from-negacyclic-matrices/144477.png","ImageObject",300,407,{"name":92,"@type":93},"Lucas Vance","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-17","2026-08-26",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",11,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What are the main goals of the paper?","Question",{"text":113,"@type":114},"The paper uses negacyclic matrices to construct orthogonal designs and thereby obtain Hadamard matrices. It also establishes results on what parameter tuples are possible for specific orders.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How does the paper define an orthogonal design?",{"text":118,"@type":114},"An orthogonal design of order n, written OD(n; s1, s2, ..., sk), is defined by a matrix A whose entries satisfy a specified type condition in terms of variables x1, ..., xk.",{"name":120,"@type":111,"acceptedAnswer":121},"What is a negashift (negacyclic) matrix in this work?",{"text":122,"@type":114},"The paper defines negashift matrices as polynomials in a negashift matrix N. It then studies algebraic relations between matrices formed from polynomials in N.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},144477,1787703031,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":41},549768064622,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","Orthogonal Designs from Nega􀁣y􀁣li􀁣 Matri􀁣es  \nKen Finlayson and Jennifer Seberry  \nS􀁣ho ol of IT and Computer S􀁣ien􀁣e  \nUniversity of Wollongong  \nNSW 2522  \nAustralia  \n19 Mar􀁣h 2003  \nAbstra􀁣t  \nWe study the use of nega􀁣y􀁣li􀁣 matri􀁣es to form orthogonal designs and hen􀁣e Hadamard matri􀁣es. We give results for all possible tuple for order  \n12, all but 3 for order 20 and all but 3 for order 28.  \nKey words and phrases: Orthogonal designs, nega􀁣y􀁣li matri􀁣es, Hadamard matri􀁣es, variant Goethals-Seidel array  \nAMS Sub je􀁣t Classi􀀌􀁣ation:  \n1 Intro du􀁣tion  \nAn orthogonal design of order n and denoted OD ( n ; s1 ; s2 ; : : : ; sk), is af0; 􀀆x1 ; 􀀆x2 ; : : : ; 􀀆xkg satisfying  \ntype (s1 ; s2 ; : matrix A of  \n: : ; sk ) in order n  \nvariables x1 ; x2 ; : : : ; xk , with entries in the set  \nk AAT = ( X  \ni=1  \nsix2i) In ;  \nwhere In is the identity matrix of order n. Alternatively, the rows of A are formally orthogonal and ea􀁣h row has pre􀁣isely si entries of the type 􀀆xi . In [1 ℄, where this was 􀀌rst de􀀌ned, it was mentioned that  \nk ATA = ( X  \ni=1  \nsix2i) In  \nand so our alternative des􀁣ription of A applies equally well to the 􀁣olumns of A.  \nAn Hadamard matrix H of order n is a square (1 ; 􀀀1) matrix having inner pro du􀁣t of distin􀁣t rows zero. Hen􀁣e H HT = nIn . We note that n = 1 ; 2 or n 􀀑 0 (mod 4) .  \nA matrix A 􀀆 I is skew-type ifA has zero diagonal and AT = 􀀀A. A skew-type  \nHadamard matrix is said to be skew-Hadamard.  \nCir􀁣ulant matri􀁣es of order n are polynomials in the shift matrix  \n0 0 1 0 􀀁 􀀁 􀀁 0 1  \nB 0 0 1 0 C  \nB C  \nS = B ... ... C :  \nB C  \nB 0 0 0 1 C  \n􀁀 1 0 0 0 A  \nNega􀁣y􀁣li􀁣 matri􀁣es of order n are polynomials in the negashift matrix  \n0  \nB B  \nN = B  \nB B  \n􀁀  \n0  \n0  \n.  \n.  \n.  \n0  \n􀀀1  \n1  \n0  \n0  \n0  \n0  \n1  \n0  \n0  \n􀀁 􀀁 􀀁 0  \n0  \n.  \n.  \n.  \n1  \n0  \n1  \nC C  \nC  \nC :  \nC C  \nA  \nThe ba􀁣k-diagonal matrix R of order n is the matrix whose elements rij are given by  \nrij = ( 10 if i +otherjwi=se n + 1 ;  \nwhere i; j = 1 ; : : : ; n. We note that if A; B are polynomial in S then A (BR )T =(BR )AT . We show below that the same is true if A; B are polynomial in N.  \nLastly, we de􀀌ne the nonperiodi􀁣 auto􀁣orrelation fun􀁣tion. Let  \nX = ffa11 ; : : : ; a1ng ; fa21 ; : : : ; a2ng ; : : : ; fam1 ; : : : ; amn gg  \nbe m sequen􀁣es of 􀁣ommuting variables of length n. Then the nonperiodi􀁣 auto-􀁣orrelation fun􀁣tion of the family of sequen􀁣es X (denoted NX ) is the fun􀁣tionde􀀌ned by  \nn􀀀j  \nNX (j ) = X (a1;ia1;i+j + a2;ia2;i+j + 􀀁 􀀁 􀀁 + am;iam;i+j ) :  \ni=1  \n2 Preliminary results  \nWe note some properties of the negashift matrix N given above:  \n(Ni) T = 􀀀Nn􀀀i and NiR = 􀀀RNn􀀀i :  \nHen􀁣e we have:  \nLemma 1 Ni(N jR )T = (N jR)( Ni) T .  \nProof. Ni(N jR )T = NiR(Nj)T = 􀀀NiR Nn􀀀j = NiN jR = NjNiR =  \n􀀀NjR Nn􀀀i = (N jR)( Ni) T . 2 Lemma 2 Suppose A; B are polynomial in S or N then A (BR )T = (BR )AT .  \nProof. The result for S , 􀁣ir􀁣ulant, 􀁣an be found in Wallis [6 ℄ . For N , nega􀁣y􀁣li􀁣, we note A and B are polynomials in N so by repeated appli􀁣ations of Lemma 1 we have the result. 2  \nWe now develop some properties of nega􀁣y􀁣li􀁣 matri􀁣es.  \nTheorem 1 Let Ri , Rj be two rows of a nega􀁣y􀁣li􀁣 matrix of dimension n, where 1 \u003C i \u003C j 􀀔 n . Then RiRTj = R1 Rj 􀀀i .  \nProof. Let i = 1 + s and j = 1 + s + t. If we write R1 = ( x1 ; : : : ; xn ), we have  \nR = (􀀀xn􀀀s+1 ; : : : ; 􀀀xn; x1 ; : : : ; xn􀀀s)  \n1+s  \nR1+t = (􀀀xn􀀀t+1 ; : : : ; 􀀀xn; x1 ; : : : ; xn􀀀t)  \nR1+s+t = (􀀀xn􀀀s􀀀t+1 ; : : : ; 􀀀xn; x1 ; : : : ; xn􀀀s􀀀t)  \nWe note that  \ns t  \nR z}|{ }|  \n(􀀀xn􀀀s ; : : : ; 􀀀xn ; zx1 ; : : : ; x{t; xt+1 ; : : : ; xn􀀀s)  \n=  \n1+s  \nR1+s+t = (􀀀xn􀀀s􀀀t+1 ; : : : ; 􀀀xn􀀀t ; 􀀀xn􀀀t+1 ; : : : ; 􀀀xn; x1 ; : : : ; xn􀀀s􀀀t)  \n|{z} |{z}  \ns t  \nThen  \nR RT = R RT  \ni j 1+s 1+s+t  \n= xn􀀀s+1xn􀀀s􀀀t+1 + 􀀁 􀀁 􀀁 + xn xn􀀀t 􀀀 x1xn􀀀t+1 􀀀 􀀁 􀀁 􀀁 􀀀 xt xn + xt+1 x1 + 􀀁 􀀁 􀀁 + xn􀀀s xn􀀀s􀀀t  \n= 􀀀x1xn􀀀t+1 􀀀 􀀁 􀀁 􀀁 􀀀 xt xn + xt+1 x1 + 􀀁 􀀁 􀀁 + xn􀀀s xn􀀀s􀀀t + xn􀀀s+1xn􀀀s􀀀t+1 + 􀀁 􀀁 􀀁 + xn xn􀀀t  \n= 􀀀x1xn􀀀t+1 􀀀 􀀁 􀀁 􀀁 􀀀 xt xn + xt+1 x1 + 􀀁 􀀁 􀀁 + xn xn􀀀t  \n= R1 Rt  \n= R1 Rj","cbCaib3Y2Ep7GCKt","https://ap.wps.com/l/cbCaib3Y2Ep7GCKt","pdf",165347,12,"English","# Introduction\n## Orthogonal designs and Hadamard matrices\n## Negashift and negacyclic matrices\n# Preliminary results\n## Key lemmas and properties\n## Inner product equivalence classes\n# Main theorems","[{\"question\":\"What are the main goals of the paper?\",\"answer\":\"The paper uses negacyclic matrices to construct orthogonal designs and thereby obtain Hadamard matrices. It also establishes results on what parameter tuples are possible for specific orders.\"},{\"question\":\"How does the paper define an orthogonal design?\",\"answer\":\"An orthogonal design of order n, written OD(n; s1, s2, ..., sk), is defined by a matrix A whose entries satisfy a specified type condition in terms of variables x1, ..., xk.\"},{\"question\":\"What is a negashift (negacyclic) matrix in this work?\",\"answer\":\"The paper defines negashift matrices as polynomials in a negashift matrix N. It then studies algebraic relations between matrices formed from polynomials in N.\"}]","Orthogonal Designs from Negacyclic Matrices | PDF"]