Page 263 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 263
COMBINATORIAL DESIGNS (MS-16)
pair of length = 77 in a joint paper with Turner/Bulutoglu/Geyer. In addition, we recently
contributed in a joint paper with Koutschan, several Legendre pairs of new lenghts ≡ 0 (
mod 3), as well as an algorithm that allows one to determine the full spectrum of values for the
3 -th power spectral density value. The importance of Legendre pairs lies in the fact that they
constitute a promising avenue to the Hadamard conjectrure.
Packings of Partial Difference Sets
Shuxing Li, shuxingl@sfu.ca
Simon Fraser University, Canada
As the underlying configuration behind many elegant finite structures, partial difference sets
have been intensively studied in design theory, finite geometry, coding theory, and graph theory.
Over the past three decades, there have been numerous constructions of partial difference sets
in abelian groups with high exponent, accompanied by numerous very different and delicate
techniques. Surprisingly, we manage to unify and extend a great many previous constructions
in a common framework, using only elementary methods. The key insight is that, instead of
focusing on one single partial difference set, we consider a packing of partial difference sets,
namely, a collection of disjoint partial difference sets in a finite abelian group. Although the
packing of partial difference sets has been implicitly studied in various contexts, we recognize
that a particular subgroup reveals crucial structural information about the packing. Identifying
this subgroup allows us to formulate a recursive lifting construction of packings in abelian
groups of increasing exponent.
This is joint work with Jonathan Jedwab.
Balanced Equi-n-squares
Trent Marbach, trent.marbach@gmail.com
Ryerson University, Canada
We define a d-balanced equi-n-square L = (lij), for some divisor d of n, as an n × n matrix
containing symbols from Zn in which any symbol that occurs in a row or column, occurs exactly
d times in that row or column. We show how to construct a d-balanced equi-n-square from a
partition of a Latin square of order n into d × (n/d) subrectangles. In design theory, L is
equivalent to a decomposition of Kn,n into d-regular spanning subgraphs of Kn/d,n/d. We also
study when L is diagonally cyclic, defined as when l(i+1)(j+1) = lij + 1 for all i, j ∈ Zn, which
corresponds to cyclic such decompositions of Kn,n (and thus α-labellings).
We identify necessary conditions for the existence of (a) d-balanced equi-n-squares, (b)
diagonally cyclic d-balanced equi-n-squares, and (c) Latin squares of order n which partition
into d × (n/d) subrectangles. We prove the necessary conditions are sufficient for arbitrary
fixed d ≥ 1 when n is sufficiently large, and we resolve the existence problem completely when
d ∈ {1, 2, 3}.
Along the way, we identify a bijection between α-labellings of d-regular bipartite graphs
and, what we call, d-starters: matrices with exactly one filled cell in each top-left-to-bottom-
right unbroken diagonal, and either d or 0 filled cells in each row and column. We use d-starters
to construct diagonally cyclic d-balanced equi-n-squares, but this also gives new constructions
of α-labellings.
261
pair of length = 77 in a joint paper with Turner/Bulutoglu/Geyer. In addition, we recently
contributed in a joint paper with Koutschan, several Legendre pairs of new lenghts ≡ 0 (
mod 3), as well as an algorithm that allows one to determine the full spectrum of values for the
3 -th power spectral density value. The importance of Legendre pairs lies in the fact that they
constitute a promising avenue to the Hadamard conjectrure.
Packings of Partial Difference Sets
Shuxing Li, shuxingl@sfu.ca
Simon Fraser University, Canada
As the underlying configuration behind many elegant finite structures, partial difference sets
have been intensively studied in design theory, finite geometry, coding theory, and graph theory.
Over the past three decades, there have been numerous constructions of partial difference sets
in abelian groups with high exponent, accompanied by numerous very different and delicate
techniques. Surprisingly, we manage to unify and extend a great many previous constructions
in a common framework, using only elementary methods. The key insight is that, instead of
focusing on one single partial difference set, we consider a packing of partial difference sets,
namely, a collection of disjoint partial difference sets in a finite abelian group. Although the
packing of partial difference sets has been implicitly studied in various contexts, we recognize
that a particular subgroup reveals crucial structural information about the packing. Identifying
this subgroup allows us to formulate a recursive lifting construction of packings in abelian
groups of increasing exponent.
This is joint work with Jonathan Jedwab.
Balanced Equi-n-squares
Trent Marbach, trent.marbach@gmail.com
Ryerson University, Canada
We define a d-balanced equi-n-square L = (lij), for some divisor d of n, as an n × n matrix
containing symbols from Zn in which any symbol that occurs in a row or column, occurs exactly
d times in that row or column. We show how to construct a d-balanced equi-n-square from a
partition of a Latin square of order n into d × (n/d) subrectangles. In design theory, L is
equivalent to a decomposition of Kn,n into d-regular spanning subgraphs of Kn/d,n/d. We also
study when L is diagonally cyclic, defined as when l(i+1)(j+1) = lij + 1 for all i, j ∈ Zn, which
corresponds to cyclic such decompositions of Kn,n (and thus α-labellings).
We identify necessary conditions for the existence of (a) d-balanced equi-n-squares, (b)
diagonally cyclic d-balanced equi-n-squares, and (c) Latin squares of order n which partition
into d × (n/d) subrectangles. We prove the necessary conditions are sufficient for arbitrary
fixed d ≥ 1 when n is sufficiently large, and we resolve the existence problem completely when
d ∈ {1, 2, 3}.
Along the way, we identify a bijection between α-labellings of d-regular bipartite graphs
and, what we call, d-starters: matrices with exactly one filled cell in each top-left-to-bottom-
right unbroken diagonal, and either d or 0 filled cells in each row and column. We use d-starters
to construct diagonally cyclic d-balanced equi-n-squares, but this also gives new constructions
of α-labellings.
261
