Constructing and embedding mutually orthogonal Latin squares: reviewing both new and existing results

We review results for the embedding of orthogonal partial Latin squares in orthogonal Latin squares, comparing and contrasting these with results for embedding partial Latin squares in Latin squares. We also present a new construction that uses the existence of a set of $t$ mutually orthogonal Latin squares of order $n$ to construct a set of $2t$ mutually orthogonal Latin squares of order $n^t$

On certain constructions for latin squares with no latin subsquares of order two

AbstractA latin square is said to be an N2-latin square (see[1] and [2]) if it contains no latin subsquare of order 2. The existence of N2-latin squares of all orders except 2k has been proved in [2]. Trivially, there are no such squares of orders 2 and 4. M. McLeish [3] has shown that there exist N2-latin squares of all orders 2k for k ⩾ 6. The present paper introduces a construction for N2-latin squares of all even orders n with n ≠ 0 (mod 3) and n ≠ 3 (mod 5). The problem is thus solved for the orders 24 and 25.For 24, the only remaining case, Eric Regener of the Faculty of Music, Université de Montréal, has constructed the following example of an N2-latin square and kindly granted us the permission to reproduce it here: 81234567823156784314678254682135758273461657182437458213687634512 The existence problem of N2-latin squares is thus completely solved

A Proof of the Magicness of the Siam Construction of a Magic Square

A magic square is an n x n array filled with n2 distinct positive integers 1, 2, ..., n2 such that the sum of the n integers in each row, column, and each of the main diagonals are the same. A Latin square is an n x n array consisting of n distinct symbols such that each symbol appears exactly once in each row and column of the square. Many articles dealing with the construction of magic squares introduce the Siam method as a simple\u27\u27 construction for magic squares. Rarely, however, does the article actually prove that the construction yields a magic square. In this paper, we describe how to decompose a magic square constructed by the Siam method into two orthogonal Latin squares, which in turn, leads us to a proof that the Siam construction produces a magic square

On orthogonal Latin pp-dimensional cubes

summary:We give a construction of $p$ orthogonal Latin $p$-dimensional cubes (or Latin hypercubes) of order $n$ for every natural number $n\ne 2,6$ and $p \ge 2$. Our result generalizes the well known result about orthogonal Latin squares published in 1960 by R. C. Bose, S. S. Shikhande and E. T. Parker

A lower bound on HMOLS with equal sized holes

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 fields. Feeding this into a generalized product construction, we are able to establish the lower bound $N(h^n) \ge (\log n)^{1/\delta}$ for any $\delta>2$ and all $n > n_0(h,\delta)$

Critical sets of full Latin squares

This thesis explores the properties of critical sets of the full n-Latin square and related combinatorial structures including full designs, (m,n,2)-balanced Latin rectangles and n-Latin cubes. In Chapter 3 we study known results on designs and the analogies between critical sets of the full n-Latin square and minimal defining sets of the full designs. Next in Chapter 4 we fully classify the critical sets of the full (m,n,2)-balanced Latin square, by describing the precise structures of these critical sets from the smallest to the largest. Properties of different types of critical sets of the full n-Latin square are investigated in Chapter 5. We fully classify the structure of any saturated critical set of the full n-Latin square. We show in Theorem 5.8 that such a critical set has size exactly equal to n³ - 2n² - n. In Section 5.2 we give a construction which provides an upper bound for the size of the smallest critical set of the full n-Latin square. Similarly in Section 5.4, another construction gives a lower bound for the size of the largest non-saturated critical set. We conjecture that these bounds are best possible. Using the results from Chapter 5, we obtain spectrum results on critical sets of the full n-Latin square in Chapter 6. In particular, we show that a critical set of each size between (n - 1)³ + 1 and n(n - 1)² + n - 2 exists. In Chapter 7, we turn our focus to the completability of partial k-Latin squares. The relationship between partial k-Latin squares and semi-k-Latin squares is used to show that any partial k-Latin square of order n with at most (n - 1) non-empty cells is completable. As Latin cubes generalize Latin squares, we attempt to generalize some of the results we have established on k-Latin squares so that they apply to k-Latin cubes. These results are presented in Chapter 8

Latin Squares and Their Applications to Cryptography

A latin square of order-n is an n x n array over a set of n symbols such that every symbol appears exactly once in each row and exactly once in each column. Latin squares encode features of algebraic structures. When an algebraic structure passes certain latin square tests , it is a candidate for use in the construction of cryptographic systems. A transversal of a latin square is a list of n distinct symbols, one from each row and each column. The question regarding the existence of transversals in latin squares that encode the Cayley tables of finite groups is far from being resolved and is an area of active investigation. It is known that counting the pairs of permutations over a Galois field ��pd whose point-wise sum is also a permutation is equivalent to counting the transversals of a latin square that encodes the addition group of ��pd. We survey some recent results and conjectures pertaining to latin squares and transversals. We create software tools that generate latin squares and count their transversals. We confirm previous results that cyclic latin squares of prime order-p possess the maximum transversal counts for 3 ≤ p ≤ 9. Furthermore, we create a new algorithm that uses these prime order-p cyclic latin squares as building blocks to construct super-symmetric latin squares of prime power order-pd with d \u3e 0; using this algorithm we accurately predict that super-symmetric latin squares of order-pd possess the confirmed maximum transversal counts for 3 ≤ pd ≤ 9 and the estimated lower bound on the maximum transversal counts for 9 \u3c pd ≤ 17. Also, we give some conjectures regarding the number of transversals in a super-symmetric latin square. Lastly, we use the super-symmetric latin square for the additive group of the Galois field (��32, +) to create a simplified version of Grøstl, an iterated hash function, where the compression function is built from two fixed, large, distinct permutations