Page 260 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 260
COMBINATORIAL DESIGNS (MS-16)
On the mini-symposium problem
Peter Danziger, danziger@ryerson.ca
Ryerson University, Canada
Joint work with E. Mendelsohn, B. Stevens, T. Traetta.
The Oberwolfach problem was originally stated as a seating problem:
Given v attendees at a conference with t circular tables each of which seats ai
t
people i=1 ai = v . Find a seating arrangement so that every person sits next to
each other person around a table exactly once over the r days of the conference.
The Oberwolfach problem thus asks for a decomposition of Kn (Kn − I when n is even) into
2-factors consisting of cycles with lengths a1, . . . , at.
In this talk we introduce the related mini-symposium problem, which asks for solutions to
the Oberwolfach problem on v points which contains a subsystem on m points. In the seating
context above, the larger conference contains a mini-symposium of m participants, and we also
require these m participants to be seated together for m−1 of the days.
2
We obtain a complete solution when the cycle sizes are as large as possible, i.e. m and v−m.
In addition, we provide extensive results in the case where all cycle lengths are equal, of size k
say, completely solving all cases when m | v, except possibly when k is odd and v is even. In
particular, we completely solve the case when all cycles are of length m (k = m).
A lower bound on permutation codes of distance n − 1
Peter Dukes, dukes@uvic.ca
University of Victoria, Canada
Coauthor: Sergey Bereg
A construction for mutually orthogonal latin squares (MOLS) inspired by pairwise balanced
designs is shown to hold more generally for a class of permutation codes of length n and mini-
mum distance n−1. Using ingredients when n equals a prime or a prime plus one, and applying
a number sieve, we obtain a general lower bound M (n, n − 1) ≥ n1.0797 on the size of such
codes for large n. This represents a small improvement on the guarantee given from MOLS.
Characterizing isomorphism classes of Latin squares by fractal
dimensions of image patterns
Raúl M. Falcón, rafalgan@us.es
Universidad de Sevilla, Spain
Based on the construction of pseudo-random sequences arisen from a given Latin square, Dim-
itrova and Markovski [1] described in 2007 a graphical representation of quasigroups by means
of fractal image patterns. The recognition and analysis of such patterns have recently arisen
[2,3] as an efficient new approach for classifying Latin squares into isomorphism classes. This
talk delves into this topic by focusing on the use of the differential box-counting method for
determining the mean fractal dimension of the homogenized standard sets associated to these
258
On the mini-symposium problem
Peter Danziger, danziger@ryerson.ca
Ryerson University, Canada
Joint work with E. Mendelsohn, B. Stevens, T. Traetta.
The Oberwolfach problem was originally stated as a seating problem:
Given v attendees at a conference with t circular tables each of which seats ai
t
people i=1 ai = v . Find a seating arrangement so that every person sits next to
each other person around a table exactly once over the r days of the conference.
The Oberwolfach problem thus asks for a decomposition of Kn (Kn − I when n is even) into
2-factors consisting of cycles with lengths a1, . . . , at.
In this talk we introduce the related mini-symposium problem, which asks for solutions to
the Oberwolfach problem on v points which contains a subsystem on m points. In the seating
context above, the larger conference contains a mini-symposium of m participants, and we also
require these m participants to be seated together for m−1 of the days.
2
We obtain a complete solution when the cycle sizes are as large as possible, i.e. m and v−m.
In addition, we provide extensive results in the case where all cycle lengths are equal, of size k
say, completely solving all cases when m | v, except possibly when k is odd and v is even. In
particular, we completely solve the case when all cycles are of length m (k = m).
A lower bound on permutation codes of distance n − 1
Peter Dukes, dukes@uvic.ca
University of Victoria, Canada
Coauthor: Sergey Bereg
A construction for mutually orthogonal latin squares (MOLS) inspired by pairwise balanced
designs is shown to hold more generally for a class of permutation codes of length n and mini-
mum distance n−1. Using ingredients when n equals a prime or a prime plus one, and applying
a number sieve, we obtain a general lower bound M (n, n − 1) ≥ n1.0797 on the size of such
codes for large n. This represents a small improvement on the guarantee given from MOLS.
Characterizing isomorphism classes of Latin squares by fractal
dimensions of image patterns
Raúl M. Falcón, rafalgan@us.es
Universidad de Sevilla, Spain
Based on the construction of pseudo-random sequences arisen from a given Latin square, Dim-
itrova and Markovski [1] described in 2007 a graphical representation of quasigroups by means
of fractal image patterns. The recognition and analysis of such patterns have recently arisen
[2,3] as an efficient new approach for classifying Latin squares into isomorphism classes. This
talk delves into this topic by focusing on the use of the differential box-counting method for
determining the mean fractal dimension of the homogenized standard sets associated to these
258
