[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-267273-105":59,"doc-detail-267273-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","a-satcas-method-for-enumerating-williamson-matrices-of-even-order","A SAT+CAS Method for Enumerating Williamson Matrices of Even Order","","The paper presents the first complete enumeration of Williamson matrices for even orders n\u003C45. It applies the SAT+CAS paradigm by coupling satisfiability solvers with computer algebra systems, leveraging progress in both SAT solving and symbolic computation. The study compares existence patterns and finds that Williamson matrices are generally more abundant for even orders than for odd orders, existing for every even order searched while missing for some odd orders.",{"@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/a-satcas-method-for-enumerating-williamson-matrices-of-even-order/267273/",{"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/a-satcas-method-for-enumerating-williamson-matrices-of-even-order/267273.png","ImageObject",300,407,{"name":92,"@type":93},"Oliver","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-02","2026-09-14",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 is the main contribution of the paper?","Question",{"text":113,"@type":114},"It provides the first complete enumeration of Williamson matrices for even orders n\u003C45.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How does the SAT+CAS method work in this context?",{"text":118,"@type":114},"It couples SAT solvers with a computer algebra system so symbolic computation and satisfiability checking jointly drive the enumeration.",{"name":120,"@type":111,"acceptedAnswer":121},"What existence pattern is observed between even and odd orders?",{"text":122,"@type":114},"Williamson matrices tend to be more abundant in even orders; they exist for every even order searched, while they do not exist for some odd orders.","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},267273,1789416586,{"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":36,"preview_page_count":4},8796095461610,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","A SAT+CAS Method for Enumerating Williamson Matrices of Even Order  \nCurtis Bright 1 , Ilias Kotsireas2 , and Vijay Ganesh 1  \n1 University of Waterloo  \n2 Wilfrid Laurier University  \nfcbright; ikotsire;[vganesh](vganeshg@uwaterloo.ca)[g](vganeshg@uwaterloo.ca)[@uwaterloo.ca](vganeshg@uwaterloo.ca)  \nAbstract. We present for the first time a complete enumeration of Williamson matrices of even order n \u003C 45. The enumeration method relies on the recently proposed SAT+CAS paradigm of coupling SAT solvers with computer algebra systems, thereby taking advantage of the advances made in both the field of satisfiability checking and the field of symbolic computation. Our results show that Williamson matrices of even order tend to be more abundant than those of odd orders; such matrices exist for every even order in which we performed a search but they do not exist for some odd orders.  \n1 Introduction  \nIn 1944, the mathematician John Williamson introduced the type of matrix which now bear his name in the process of studying the Hadamard conjecture from combinatorial design theory [20] . This conjecture states that Hadamard matrices—matrices H 2 f􀀆1gn􀀂n which satisfy HHT = nIn—exist for all orders n divisible by 4. Williamson defined a new collection of matrices which have since been extensively used to construct Hadamard matrices of many different orders, including some orders for which no other method has been able to work successfully. Williamson matrices have also been studied for their application to long-range digital communication systems and this motivated mathematicians from NASA's Jet Propulsion Laboratory to construct Williamson matrices of order 23 in 1961 [5] .  \nAlthough Williamson defined his matrices for both even and odd orders, the even case has only been studied quite recently [7], though generalizations of Williamson matrices (sometimes referred to as Williamson-type matrices) have formerly been studied in even orders [18] . The algorithms used for enumerating Williamson matrices prior to 2016 (e.g., [4, 13 , 16]) rely on properties only available in odd orders making it impossible to use them for enumerating Williamson matrices of even order. In light of this, it is interesting to develop algorithms for enumerating Williamson matrices which also work for even orders.  \nUnfortunately, it would not be possible to resolve the Hadamard conjecture by only studying Williamson matrices of even order, since Williamson matrices of even order can only be used to construct Hadamard matrices of orders which are divisible by 8. However, it is still not even known if Hadamard matrices exist  \nfor all orders divisible by 8, so nevertheless studying Williamson matrices of even order has the potential to shed light on the Hadamard conjecture as well.  \nOn the other hand, if it was possible to show that Williamson matrices exist for all odd orders this would resolve the Hadamard conjecture, leading to the related conjecture that Williamson matrices exist in all odd orders. As Richard Turyn wrote [17]:  \nIt has been conjectured that an Hadamard matrix of this [Williamson] type might exist of every order 4t, at least for t odd.  \nHowever, this conjecture was shown to be false by the mathematician Dragomir Ðoković who showed that such matrices do not exist in order t = 35 [16] . Later, Holzmann, Kharaghani, and Tayfeh-Rezaie [13] showed that Williamson matrices also do not exist for orders 47 , 53, and 59 but exist for all other odd orders under 60.  \nIn this paper we provide for the first time3 an enumeration of Williamson matrices in orders which are not odd. In doing so, we uncover an interesting but so far unexplained phenomenon that there tend to be more Williamson matrices in even orders than there are in odd orders. In fact, Williamson matrices exist for every even order in which we performed a search. In light of this, Turyn's remark stating that the Williamson conjecture should apply “at least for t odd”seems unnecessary. This leads ","cbCaipJ4D4ReAE50","https://ap.wps.com/l/cbCaipJ4D4ReAE50","pdf",431126,16,"English","# Abstract\n# Introduction\n## Background and motivation\n## Updated Williamson conjecture\n# Background\n## Williamson matrices\n# Method\n## SAT+CAS paradigm\n# Results\n## Enumeration for n \u003C 45","[{\"question\":\"What is the main contribution of the paper?\",\"answer\":\"It provides the first complete enumeration of Williamson matrices for even orders n\\u003c45.\"},{\"question\":\"How does the SAT+CAS method work in this context?\",\"answer\":\"It couples SAT solvers with a computer algebra system so symbolic computation and satisfiability checking jointly drive the enumeration.\"},{\"question\":\"What existence pattern is observed between even and odd orders?\",\"answer\":\"Williamson matrices tend to be more abundant in even orders; they exist for every even order searched, while they do not exist for some odd orders.\"}]","A SAT+CAS Method for Enumerating Williamson Matrices of Even Order | PDF"]