一对正交的不完全拉丁方完备大集

不完全拉丁方完备大集,记作LDILS⁺(n+a,a),由两两不交的n个不完全拉丁方ILS(n+a,a)和a个拉丁方LS(n)构成.正交的不完全拉丁方完备大集,记作OLDILS⁺(n+a,a),由一对正交的LDILS⁺(n+a,a)构成.本文研究OLDILS⁺(n+a,a)的存在性问题,利用有限域上的直接构造以及引入辅助设计OLS_n⁺(n)进行积构造,得到若干OLDILS⁺(n+a,a)的无穷类.     

自正交拉丁方存在性的一个简短证明

本文是欧拉猜想的一个简明反证.文中给出了所有n≠2,3,6阶的一族自正交拉丁方. §1.引言 1959年和1960年,Bose,Shrikhande和Parker反证了欧拉关于不存在4t+2阶正交拉丁方的猜想,解决了正交拉丁方的存在性问题.1973年Brayton,Coppersmith     

正交对角拉丁方的构造

对角拉丁方是主对角线和反对角线均为截态的拉丁方。本文推广了朱烈［１］中关于正交对角拉丁方的主要构作。作为这些构作的应用，作者改进了Ｗａｌｌｉｓ和朱烈［１，２］中关于正交对角拉丁方的结果。     

关于s阶的两两正交拉丁方的最大数目——筛法的应用——

<正> §1.序言本文的目的为给出[1]中宣布的结果的详细证明.本文还略为改良了这些结果.由 s 个相异元素(例如1,2,…,s)构成的 s×s 方阵,如果每一元素都在方阵的任何一行与任何一列中出现一次,而且恰好出现一次,则称这种方阵为 s 阶的拉丁方.又若将两个,阶的拉丁方重选在一起,则上面拉丁方的任何元素都正好遇见下面拉丁方的每一元素一次,而且恰好一次,就称这两个拉丁方是正交的.     

正交拉丁方组和泛对角线幻方的一种构造法

利用线性取余变换构造素数阶完备正交拉丁方组,给出泛对角线幻方的一种构造法.     

强对称的自正交对角拉丁方

1 引言 一个ｎ阶拉丁方是含ｎ个相异元素的集合Ｎ上的一个ｎ阶方阵，其每一行和每一列都是Ｎ的一个置换．ｎ阶拉丁方的一条截态是位于不同行不同列的ｎ个位置使得其中的ｎ个元素两两相异．ｎ阶对角拉丁方是一个ｎ阶拉丁方，其主对角线（位置（）与反对角线（位置（）均为截态． 两个ｎ阶拉丁方Ａ和Ｂ称为正交的（简记作Ａ上Ｂ），如果把它们迭合在一起时，拉丁方Ａ的每一个记号与拉丁方Ｂ的每一个记号相遇一次且仅相遇一次．如果一个ｎ阶拉丁方Ｌ和它自己的转置正交，则称Ｌ为一个自正交的拉丁方，简记为ＳＯＬＳ（ｎ）． ｎ阶自正交对角拉…     

基于三种重复方式下超希腊拉丁方设计的方差分析

超希腊拉丁方设计可以从多个不同方向控制试验的误差,在医学和工农业生产等领域有着广泛的应用.因此,对超希腊拉丁方设计进行方差分析有着重要意义.文章首先给出了超希腊拉丁方设计的方差分析,并讨论了在三种不同重复方式下的超希腊拉丁方设计的方差分析,最后通过例子验证所获的理论结果.     

带两洞的不完全自正交拉丁方的存在性

讨论不完全自正交拉丁方ISOLS(v;3,3)的存在性问题.证明当v≥12,v{13,14,15,16,17,18,19,20,21,22,23,24,25,27,28,29,30,31,33,35,36}时,存在ISOLS(v;3,3).     

PBD闭集的有限生成集

本文给出了ａ＝７，８时ＰＢＤ闭集Ｈａ的有限生成集和Ｈ＾０６＝（ｖ：ｖ≡０，１（ｍｏｄ６）的有限生成集。     

非负矩阵Hadamard积的最大特征值的新上界

关于非负矩阵A和B的Hadamard积的最大特征值的上界问题,主要利用Gerschgorin定理和Brauer定理给出了新的估计式,并把新结果与现有结果进行了比较.数值算例表明新结果在只依赖矩阵元素的条件下改进了现有的一些估计式.     

New quasi-symmetric designs constructed using mutually orthogonal Latin squares and Hadamard matrices

Using Hadamard matrices and mutually orthogonal Latin squares, we construct two new quasi-symmetric designs, with parameters 2 − (66,30,29) and 2 − (78,36,30). These are the first examples of quasi-symmetric designs with these parameters. The parameters belong to the families 2 − (2u ² − u,u ² − u,u ² − u − 1) and 2 − (2u ² + u,u ²,u ² − u), which are related to Hadamard parameters. The designs correspond to new codes meeting the Grey–Rankin bound.     

Constructing orthogonal pandiagonal Latin squares and panmagic squares from modular n‐queens solutions

In this article, we show how to construct pairs of orthogonal pandiagonal Latin squares and panmagic squares from certain types of modular n‐queens solutions. We prove that when these modular n‐queens solutions are symmetric, the panmagic squares thus constructed will be associative, where for an n × n associative magic square A = (a_ij), for all i and j it holds that a_ij + an?i?1,n?j?1 = c for a fixed c. We further show how to construct orthogonal Latin squares whose modular difference diagonals are Latin from any modular n‐queens solution. As well, we analyze constructing orthogonal pandiagonal Latin squares from particular classes of non‐linear modular n‐queens solutions. These pandiagonal Latin squares are not row cyclic, giving a partial solution to a problem of Hedayat. © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 221–234, 2007     

Atomic Latin squares of order eleven

A Latin square is pan‐Hamiltonian if the permutation which defines row i relative to row j consists of a single cycle for every i ≠ j. A Latin square is atomic if all of its conjugates are pan‐Hamiltonian. We give a complete enumeration of atomic squares for order 11, the smallest order for which there are examples distinct from the cyclic group. We find that there are seven main classes, including the three that were previously known. A perfect 1‐factorization of a graph is a decomposition of that graph into matchings such that the union of any two matchings is a Hamiltonian cycle. Each pan‐Hamiltonian Latin square of order n describes a perfect 1‐factorization of K_n,n, and vice versa. Perfect 1‐factorizations of K_n,n can be constructed from a perfect 1‐factorization of K_n+1. Six of the seven main classes of atomic squares of order 11 can be obtained in this way. For each atomic square of order 11, we find the largest set of Mutually Orthogonal Latin Squares (MOLS) involving that square. We discuss algorithms for counting orthogonal mates, and discover the number of orthogonal mates possessed by the cyclic squares of orders up to 11 and by Parker's famous turn‐square. We find that the number of atomic orthogonal mates possessed by a Latin square is not a main class invariant. We also define a new sort of Latin square, called a pairing square, which is mapped to its transpose by an involution acting on the symbols. We show that pairing squares are often orthogonal mates for symmetric Latin squares. Finally, we discover connections between our atomic squares and Franklin's diagonally cyclic self‐orthogonal squares, and we correct a theorem of Longyear which uses tactical representations to identify self‐orthogonal Latin squares in the same main class as a given Latin square. © 2003 Wiley Periodicals, Inc.     

On the orthogonal Latin squares polytope

Since 1782, when Euler addressed the question of existence of a pair of orthogonal Latin squares (OLS) by stating his famous conjecture, these structures have remained an active area of research. In this paper, we examine the polyhedral aspects of OLS. In particular, we establish the dimension of the OLS polytope, describe all cliques of the underlying intersection graph and categorize them into three classes. Two of these classes are shown to induce facet-defining inequalities of Chvátal rank two. For each such class, we provide a polynomial separation algorithm of the lowest possible complexity.     

A tripling construction for mutually orthogonal symmetric hamiltonian double Latin squares

We provide two new constructions for pairs of mutually orthogonal symmetric hamiltonian double Latin squares. The first is a tripling construction, and the second is derived from known constructions of hamilton cycle decompositions of when is prime.     

Sets of orthogonal hypercubes of class r

A (d,n,r,t)-hypercube is an n×n×?×n (d-times) array on nr symbols such that when fixing t coordinates of the hypercube (and running across the remaining d−t coordinates) each symbol is repeated nd−r−t times. We introduce a new parameter, r, representing the class of the hypercube. When r=1, this provides the usual definition of a hypercube and when d=2 and r=t=1 these hypercubes are Latin squares. If d?2r, then the notion of orthogonality is also inherited from the usual definition of hypercubes. This work deals with constructions of class r hypercubes and presents bounds on the number of mutually orthogonal class r hypercubes. We also give constructions of sets of mutually orthogonal hypercubes when n is a prime power.