The Search for Pseudo Orthogonal Latin Squares of Order Six

We report on the completecomputer search for a strongly regular graph with parameters(36,15,6,6) and chromatic number six. The resultis that no such graph exists.     

Generating spherical Eulerian triangulations

We show that all spherical Eulerian triangulations can be inductively generated from the set of all even double wheels using just one of the two local transformations used in the algorithm by Brinkmann and McKay and originally proposed by Batagelj.     

用正交对角拉丁方及幻矩与和阵构作mn阶标准二次幻方

当m和n为同奇或同偶的正整数且m,n≠1,2,3,6时,用m和n阶正交对角拉丁方及{0,1,…,mn-1）上的m×n幻矩与和阵,构作了mn阶标准二次幻方．     

Properties, isomorphisms and enumeration of 2-Quasi-Magic Sudoku grids

A Sudoku grid is a 9×9 Latin square further constrained to have nine non-overlapping 3×3 mini-grids each of which contains the values 1–9. In Δ-Quasi-Magic Sudoku a further constraint is imposed such that every row, column and diagonal in each mini-grid sums to an integer in the interval [15−Δ,15+Δ]. The problem of proving certain (computationally known) results for Δ=2 concerning mini-grids and bands (rows of mini-grids) was posed at the British Combinatorial Conference in 2007. These proofs are presented and extensions of these provide a full combinatorial enumeration for the total number of completed 2-Quasi-Magic Sudoku grids. It is also shown that there are 40 isomorphism classes of completed 2-Quasi-Magic Sudoku grids.     

Negative Latin square type partial difference sets in nonabelian groups of order 64

There exist few examples of negative Latin square type partial difference sets (NLST PDSs) in nonabelian groups. We present a list of 176 inequivalent NLST PDSs in 48 nonisomorphic, nonabelian groups of order 64. These NLST PDSs form 8 nonisomorphic strongly regular graphs. These PDSs were constructed using a combination of theoretical techniques and computer search, both of which are described. The search was run exhaustively on 212/267 nonisomorphic groups of order 64.     

一种OLS/OOC二维光正交码的构造与性能分析

利用不同的序列作为波长跳频序列和时间扩频序列可以构造出不同的二维光正交码在众多文献中已有所报道.在经过正交拉丁方(OLS)与跳频序列的相关性研究之后.做了以下主要工作:首先,将正交拉丁方(OLS)序列作为波长跳频序列,结合一维时间扩频序列(OOC),构造了一种OLS/OOC二维光正交码.然后,本文对构造的OLS/OOC进行了多种性能仿真和分析.相对于PC/OOC、OCFHC/OOC等二维光正交码而言,OLS/OOC的波长数并不局限于素数,更能充分利用MWOCDMA系统中的有效波长数.仿真和分析表明:码字具有很好的相关性能,码字容量直逼理论极限,为一种渐近最优二维光正交码.     

Latin squares with bounded size of row prefix intersections

A latin square is a matrix of size n×n with entries from the set {1,…,n}, such that each row and each column is a permutation on {1,…,n}. We show how to construct a latin square such that for any two distinct rows, the prefixes of length h of the two rows share at most about h²/n elements. This upper bound is close to optimal when contrasted with a lower bound derived from the Second Johnson bound [6].     

Some new HSOLSSOMs of types hn and 1m u1

In this paper, two new direct construction methods are given for holey self‐orthogonal Latin squares with a symmetric orthogonal mate (HSOLSSOMs). Some new HSOLSSOMs using already known methods are also given. The known existence results for HSOLSSOMs of types 1^m u¹ and hⁿ are improved; for type 1^m u¹ there remain just four possible exceptions with u odd and 3 ≤ u ≤ 15; for type hⁿ, there are just two possible exceptions remaining, for (h, n) = (6, 12) and (6, 18). As a byproduct, the known existence results for three holey mutually orthogonal Latin squares (3 HMOLS) are also improved. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 435–444, 2001     

最佳拉丁方与高级原幻方

本文证明了 (n ,2 ) =(n ,3) =1时 ,有n阶的正交的最佳拉丁方。若n =4k ,或n是个不为 3的奇数 ,则有n阶的正交的高级原幻方