Combinatorial Constructions For T-Multimagic Squares

Magic square is one of the research objects of combinatorial designs. It has been usedin quantum information, digital halftoning, encryption technology, etc. The t-multimagicsquares is an important topic in magic square. In this paper, the constructions and existenceof t-multimagic squares are investigated.For t=1, a magic square with strong property called symmetrical pandiagonal elemen-tary magic square is investigated. A symmetrical pandiagonal elementary magic square canbe obtained by using a pair of strongly symmetrical and weakly pandiagonal orthogonal Latinsquares. Center-complementary row magic rectangles are used to give the constructions ofstrongly symmetrical and weakly pandiagonal orthogonal Latin squares of odd order. Theproduct construction of orthogonal Latin squares is generalized to obtain the basic construc-tion of strongly symmetrical and weakly pandiagonal orthogonal Latin squares of even order.It is proved that there exists a symmetrical pandiagonal elementary magic square if and onlyif n>4and n≡0,1,3(mod4) with one possible exception n=12.For t≥2, the large set of orthogonal arrays (LOA) is used. The concepts of doublelarge set of orthogonal arrays (DLOA) and strong double large set of orthogonal arrays (SD-LOA) are introduced. The relationships are established between LOA and row t-multimagicrectangle, DLOA and t-multimagic rectangle, SDLOA and t-multimagic square, respectively.Constructions of DLOA and SDLOA over finite field are investigated and families of DLOAand SDLOA are obtained. Orthogonal diagonal Latin squares are used to give the existenceof the SDLOAs of strength2and constraint4. Further, the concepts of complementary t-multimagic rectngles and complementary t-multimagic squares are introduced and construc-tions of t-multimagic rectngles and t-multimagic squares are obtained. The orthogonal diago-nal Latin squares, Kotzig arrays and SDLOA are used to give some families of complementaryt-multimagic rectngles and complementary t-multimagic squares. Consequently, some fami-lies of t-multimagic rectngles and t-multimagic squares are obtained.The paper consists of six chapters and they are organized as follows. In Chapter1, the research background of the full paper, the related concepts, the knownresults and our main results are presented.In Chapter2, some constructions of strongly symmetrical and weakly pandiagonal or-thogonal Latin squares are offered and it is proved that there exists a symmetrical pandiagonalelementary magic square if and only if n>4and n≡0,1,3(mod4) with one possibleexception n=12.In Chapter3, t-multimagic rectangles are investigated. Firstly, The concept of DLOAis introduced, a construction of t-multimagic rectngle based on DLOA is given and a con-struction of DLOA of prime power order is presented. Then the concept of complementaryt-multimagic rectngle is introduced and a recursive construction of t-multimagic rectngle isgiven. Finally, a special Kotzig array is used and hence two families of t-multimagic rectngleare given.In Chapter4, t-multimagic squares are investigated. The concept of SDLOA is intro-duced and a construction of t-multimagic squares based on SDLOA is given. Then the exis-tence of SDLOA with strength2and constraints4is given and the existence of SDLOA ofprime power order is also presented. As their applications, two families t-multimagic squaresare obtained.In Chapter5, recursive constructions of t-multimagic squares are investigated. The prod-uct construction of t-multimagic squares is proved and complementary t-multimagic rectnglesare introduced. Some recursive constructions of t-multimagic squares are given and someclasses of t-multimagic squares are obtained by using diagonal orthogonal Latin squares,Kotzig array and SDLOA.In Chapter6, some problems for further research are listed.