Page 262 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 262
COMBINATORIAL DESIGNS (MS-16)
is completable to a (n, k, 1)-design provided that n is sufficiently large and obeys the obvious
necessary conditions for an (n, k, 1)-design to exist. This result is sharp for all k. I will also
mention some related results concerning edge decompositions of almost complete graphs into
copies of Kk.
Balancedly splittable orthogonal designs
Hadi Kharaghani, kharaghani@uleth.ca
University of Lethbridge, Canada
Coauthors: Thomas Pender, Sho Suda
As an extension of balancedly splittable Hadamard matrices, the concept of balancedly split-
table orthogonal designs is introduced along with a recursive construction. Among the other
results, a class of equiangular tight frames over the real, complex, and quaternions meeting the
Delsarte-Goethals-Seidel upper bound is obtained.
Resolving sets and identifying codes in finite geometries
György Kiss, kissgy@cs.elte.hu
Eötvös Loránd University, Budapest, Hungary, and University of Primorska, Koper, Slovenia
Let Γ = (V, E) be a finite, simple, undirected graph. A vertex v ∈ V is resolved by S =
{v1, . . . , vn} ⊂ V if the list of distances (d(v, v1), d(v, v2), . . . , d(v, vn)) is unique. S is a
resolving set for Γ if it resolves all the elements of V .
A subset D ⊂ V is a dominating set if each vertex is either in D or adjacent to a vertex in D.
A vertex s separates u and v if exactly one of u and v is in N [s]. A subset S ⊂ V is a separating
set if it separates every pair of vertices of G. Finally, a subset C ⊂ V is an identifying code for
V if it is both a dominating and separating set.
In this talk resolving sets and identifying codes for graphs arising from finite geometries
(e.g. Levi graphs of projective and affine planes and spaces, generalized quadrangles) are con-
sidered. We present several constructions and give estimates on the sizes of these objects.
20 years of Legendre Pairs
Ilias Kotsireas, ikotsire@gmail.com
WLU, Canada
Legendre pairs were introduced in 2001 by Seberry and her students, as a means to construct
Hadamard matrices via a two-circulant core construction. A Legendre pair consists of two
sequences of odd length , with elements from {−1, +1}, such that their respective autocorrela-
tion coefficients sum to −2, or (equivalently) their respective power spectral density coefficients
sum to 2 + 2. Legendre pairs of every odd prime length exist, via a simple construction using
the Legendre symbol. We will review known constructions for Legendre pairs. We will discuss
various results on Legendre pairs during the past 20 years, including the concept of compres-
sion, introduced in a joint paper with Djokovic, as well as the computational state-of-the-art of
the search for Legendre pairs. In particular, we recently contributed the only known Legendre
260
is completable to a (n, k, 1)-design provided that n is sufficiently large and obeys the obvious
necessary conditions for an (n, k, 1)-design to exist. This result is sharp for all k. I will also
mention some related results concerning edge decompositions of almost complete graphs into
copies of Kk.
Balancedly splittable orthogonal designs
Hadi Kharaghani, kharaghani@uleth.ca
University of Lethbridge, Canada
Coauthors: Thomas Pender, Sho Suda
As an extension of balancedly splittable Hadamard matrices, the concept of balancedly split-
table orthogonal designs is introduced along with a recursive construction. Among the other
results, a class of equiangular tight frames over the real, complex, and quaternions meeting the
Delsarte-Goethals-Seidel upper bound is obtained.
Resolving sets and identifying codes in finite geometries
György Kiss, kissgy@cs.elte.hu
Eötvös Loránd University, Budapest, Hungary, and University of Primorska, Koper, Slovenia
Let Γ = (V, E) be a finite, simple, undirected graph. A vertex v ∈ V is resolved by S =
{v1, . . . , vn} ⊂ V if the list of distances (d(v, v1), d(v, v2), . . . , d(v, vn)) is unique. S is a
resolving set for Γ if it resolves all the elements of V .
A subset D ⊂ V is a dominating set if each vertex is either in D or adjacent to a vertex in D.
A vertex s separates u and v if exactly one of u and v is in N [s]. A subset S ⊂ V is a separating
set if it separates every pair of vertices of G. Finally, a subset C ⊂ V is an identifying code for
V if it is both a dominating and separating set.
In this talk resolving sets and identifying codes for graphs arising from finite geometries
(e.g. Levi graphs of projective and affine planes and spaces, generalized quadrangles) are con-
sidered. We present several constructions and give estimates on the sizes of these objects.
20 years of Legendre Pairs
Ilias Kotsireas, ikotsire@gmail.com
WLU, Canada
Legendre pairs were introduced in 2001 by Seberry and her students, as a means to construct
Hadamard matrices via a two-circulant core construction. A Legendre pair consists of two
sequences of odd length , with elements from {−1, +1}, such that their respective autocorrela-
tion coefficients sum to −2, or (equivalently) their respective power spectral density coefficients
sum to 2 + 2. Legendre pairs of every odd prime length exist, via a simple construction using
the Legendre symbol. We will review known constructions for Legendre pairs. We will discuss
various results on Legendre pairs during the past 20 years, including the concept of compres-
sion, introduced in a joint paper with Djokovic, as well as the computational state-of-the-art of
the search for Legendre pairs. In particular, we recently contributed the only known Legendre
260
