98,323 research outputs found
Difference Covering Arrays and Pseudo-Orthogonal Latin Squares
Difference arrays are used in applications such as software testing,
authentication codes and data compression. Pseudo-orthogonal Latin squares are
used in experimental designs. A special class of pseudo-orthogonal Latin
squares are the mutually nearly orthogonal Latin squares (MNOLS) first
discussed in 2002, with general constructions given in 2007. In this paper we
develop row complete MNOLS from difference covering arrays. We will use this
connection to settle the spectrum question for sets of 3 mutually
pseudo-orthogonal Latin squares of even order, for all but the order 146
Parity of Sets of Mutually Orthogonal Latin Squares
Every Latin square has three attributes that can be even or odd, but any two
of these attributes determines the third. Hence the parity of a Latin square
has an information content of 2 bits. We extend the definition of parity from
Latin squares to sets of mutually orthogonal Latin squares (MOLS) and the
corresponding orthogonal arrays (OA). Suppose the parity of an
has an information content of bits. We show that
. For the case corresponding to projective
planes we prove a tighter bound, namely when
is odd and when is even. Using the
existence of MOLS with subMOLS, we prove that if
then for all sufficiently large .
Let the ensemble of an be the set of Latin squares derived by
interpreting any three columns of the OA as a Latin square. We demonstrate many
restrictions on the number of Latin squares of each parity that the ensemble of
an can contain. These restrictions depend on and
give some insight as to why it is harder to build projective planes of order than for . For example, we prove that when it is impossible to build an for which all
Latin squares in the ensemble are isotopic (equivalent to each other up to
permutation of the rows, columns and symbols)
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
Enumerating extensions of mutually orthogonal Latin squares
Two n×n Latin squares L1,L2 are said to be orthogonal if, for every ordered pair (x, y) of symbols, there are coordinates (i, j) such that L1(i,j)=x and L2(i,j)=y. A k-MOLS is a sequence of k pairwise-orthogonal Latin squares, and the existence and enumeration of these objects has attracted a great deal of attention. Recent work of Keevash and Luria provides, for all fixed k, log-asymptotically tight bounds on the number of k-MOLS. To study the situation when k grows with n, we bound the number of ways a k-MOLS can be extended to a (k+1)-MOLS. These bounds are again tight for constant k, and allow us to deduce upper bounds on the total number of k-MOLS for all k. These bounds are close to tight even for k linear in n, and readily generalise to the broader class of gerechte designs, which include Sudoku squares
Critical Sets in Latin Squares and Associated Structures
A critical set in a Latin square of order n is a set of entries in an n×n array which can be embedded in precisely one Latin square of order n, with the property that if any entry of the critical set is deleted, the remaining set can be embedded in more than one Latin square of order n. The number of critical sets grows super-exponentially as the order of the Latin square increases. It is difficult to find patterns in Latin squares of small order (order 5 or less) which can be generalised in the process of creating new theorems. Thus, I have written many algorithms to find critical sets with various properties in Latin squares of order greater than 5, and to deal with other related structures. Some algorithms used in the body of the thesis are presented in Chapter 3; results which arise from the computational studies and observations of the patterns and subsequent results are presented in Chapters 4, 5, 6, 7 and 8. The cardinality of the largest critical set in any Latin square of order n is denoted by lcs(n). In 1978 Curran and van Rees proved that lcs(n)=2 and α≠3), and a new lower bound on lcs(4m). It also discusses critical sets in intercalate-rich Latin squares of orders 11 and 14. In Chapter 6 a construction is given which verifies the existence of a critical set of size n²÷ 4 + 1 when n is even and n>=6. The construction is based on the discovery of a critical set of size 17 for a Latin square of order 8. In Chapter 7 the representation of Steiner trades of volume less than or equal to nine is examined. Computational results are used to identify those trades for which the associated partial Latin square can be decomposed into six disjoint Latin interchanges. Chapter 8 focusses on critical sets in Latin squares of order at most six and extensive computational routines are used to identify all the critical sets of different sizes in these Latin squares
Parity of transversals of Latin squares
We introduce a notion of parity for transversals, and use it to show that in
Latin squares of order , the number of transversals is a multiple of
4. We also demonstrate a number of relationships (mostly congruences modulo 4)
involving , where is the number of diagonals of a given
Latin square that contain exactly different symbols.
Let denote the matrix obtained by deleting row and column
from a parent matrix . Define to be the number of transversals
in , for some fixed Latin square . We show that for all and . Also, if has odd order then the
number of transversals of equals mod 2. We conjecture that for all .
In the course of our investigations we prove several results that could be of
interest in other contexts. For example, we show that the number of perfect
matchings in a -regular bipartite graph on vertices is divisible by
when is odd and . We also show that for all , when is an integer matrix of odd
order with all row and columns sums equal to
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