Mutually orthogonal frequency squares (MOFS) of type $F(mλ;λ)$ generalize the structure of mutually orthogonal Latin squares: rather than each of $m$ symbols appearing exactly once in each row and in each column of each square, the repetition number is $λ\ge 1$. A classical upper bound for the number of such MOFS is $\frac{(mλ-1)^2}{m-1}$. We introduce a new representation of MOFS of type $F(mλ;λ)$, as a linear combination of $\{0,1\}$ arrays. We use this representation to give an elementary proof of the classical upper bound, together with a structural constraint on a set of MOFS achieving the upper bound. We then use this representation to establish a maximality criterion for a set of MOFS of type $F(mλ;λ)$ when $m$ is even and $λ$ is odd, which simplifies and extends a previous analysis [T. Britz, N.J. Cavenagh, A. Mammoliti, I.M. Wanless, Mutually orthogonal binary frequency squares, Electron. J. Combin., 27(#P3.7), 2020, 26 pages] of the case when $m=2$ and $λ$ is odd.
We introduce a graph attached to mutually orthogonal Sudoku Latin squares. The spectra of the graphs obtained from finite fields are explicitly determined. As a corollary, we then use the eigenvalues to distinguish non-isomorphic Sudoku Latin squares.
An ${\ell}$-cycle system ${\mathcal F}$ of a graph $Γ$ is a set of ${\ell}$-cycles which partition the edge set of $Γ$. Two such cycle systems ${\mathcal F}$ and ${\mathcal F}'$ are said to be {\em orthogonal} if no two distinct cycles from ${\mathcal F}\cup {\mathcal F}'$ share more than one edge. Orthogonal cycle systems naturally arise from face $2$-colourable polyehdra and in higher genus from Heffter arrays with certain orderings. A set of pairwise orthogonal $\ell$-cycle systems of $Γ$ is said to be a set of mutually orthogonal cycle systems of $Γ$. Let $μ(\ell,n)$ (respectively, $μ'(\el…
A frequency rectangle of type FR$(m,n;q)$ is an $m \times n$ matrix such that each symbol from a set of size $q$ appears $n/q$ times in each row and $m/q$ times in each column. Two frequency rectangles of the same type are said to be orthogonal if, upon superimposition, each possible ordered pair of symbols appear the same number of times. A set of $k$ frequency rectangles in which every pair is orthogonal is called a set of mutually orthogonal frequency rectangles, denoted by $k$--MOFR$(m,n;q)$. We show that a $k$--MOFR$(2,2n;2)$ and an orthogonal array OA$(2n,k,2,2)$ are equivalent. We also …
Correlation immune Boolean functions play an important role in the implementation of efficient masking countermeasures for side-channel attacks in cryptography. In this paper, we investigate a method to construct correlation immune functions through families of mutually orthogonal cellular automata (MOCA). First, we show that the orthogonal array (OA) associated to a family of MOCA can be expanded to a binary OA of strength at least 2. To prove this result, we exploit the characterization of MOCA in terms of orthogonal labelings on de Bruijn graphs. Then, we use the resulting binary OA to defi…
Carleton University（卡尔顿大学）；University of Windsor（温莎大学）
In this paper we provide results on using integer programming (IP) and constraint programming (CP) to search for sets of mutually orthogonal latin squares (MOLS). Both programming paradigms have previously successfully been used to search for MOLS, but solvers for IP and CP solvers have significantly improved in recent years and data on how modern IP and CP solvers perform on the MOLS problem is lacking. Using state-of-the-art solvers as black boxes we were able to quickly find pairs of MOLS (or prove their nonexistence) in all orders up to ten. Moreover, we improve the effectiveness of the so…
Choi Seok-Jeong studied Latin squares at least 60 years earlier than Euler although this was less known. He introduced a pair of orthogonal Latin squares of order 9 in his book. Interestingly, his two orthogonal non-double-diagonal Latin squares produce a magic square of order 9, whose theoretical reason was not studied. There have been a few studies on Choi's Latin squares of order 9. The most recent one is Ko-Wei Lih's construction of Choi's Latin squares of order 9 based on the two $3 \times 3$ orthogonal Latin squares. In this paper, we give a new generalization of Choi's orthogonal Latin …
A frequency square is a square matrix in which each row and column is a permutation of the same multiset of symbols. A frequency square is of type $(n;λ)$ if it contains $n/λ$ symbols, each of which occurs $λ$ times per row and $λ$ times per column. In the case when $λ=n/2$ we refer to the frequency square as binary. A set of $k$-MOFS$(n;λ)$ is a set of $k$ frequency squares of type $(n;λ)$ such that when any two of the frequency squares are superimposed, each possible ordered pair occurs equally often. A set of $k$-maxMOFS$(n;λ)$ is a set of $k$-MOFS$(n;λ)$ that is not contained in any set of…
Tokai Senior High School（东海高中）；School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学深圳数据科学学院）；Graduate School of Mathematics, Nagoya University（名古屋大学大学院数学研究科）
It is known that $N(n)$, the maximum number of mutually orthogonal latin squares of order $n$, satisfies the lower bound $N(n) \ge n^{1/14.8}$ for large $n$. For $h\ge 2$, relatively little is known about the quantity $N(h^n)$, which denotes the maximum number of `HMOLS' or mutually orthogonal latin squares having a common equipartition into $n$ holes of a fixed size $h$. We generalize a difference matrix method that had been used previously for explicit constructions of HMOLS. An estimate of R.M. Wilson on higher cyclotomic numbers guarantees our construction succeeds in suitably large finite…
On the minimum number of entries in a pair of maximal orthogonal partial Latin squares
ARC Centre of Excellence for Plant Success in Nature and Agriculture（植物在自然与农业中成功卓越中心）；School of Mathematics and Physics, University of Queensland（昆士兰大学数理学院）；School of Mathematics and Statistics, The Open University（开放大学数学与统计学院）；Department of Mathematics, Koç University（科奇大学数学系）
Myrvold's Results on Orthogonal Triples of $10 \times 10$ Latin Squares: A SAT Investigation
School of Computer Science, University of Waterloo（滑铁卢大学计算机学院）；Department of Mathematics and Statistics, University of Windsor（温莎大学数学与统计系）；School of Mathematics and Statistics, Carleton University（卡尔顿大学数学与统计学院）
