Page 265 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 265
COMBINATORIAL DESIGNS (MS-16)
and (r, λ) = 1, admitting a flag-transitive automorphism group G, mostly focusing on the
constructions of the various examples contained in it.
Testing isomorphism of circulant objects in polynomial time
Mikhael Muzychuk, misha.muzychuk@gmail.com
Ben-Gurion University of the Negev, Israel
Coauthor: Ilia Ponomarenko
We show that isomorphism testing of two cyclic combinatorial objects may be done in a poly-
nomial time of their sizes provided that both objects share the same regular cyclic group of
automorphisms given in advance.
Strictly additive 2-designs
Anamari Nakic´, anamari.nakic@fer.hr
University of Zagreb, Croatia
Coauthor: Marco Buratti
This work draws inspiration from an interesting theory developed in [1]. A design (V, B) is said
to be additive if V is a subset of an abelian group G and the elements of any block B ∈ B sum
up to zero. We propose to speak of a strictly additive design when V coincides with G.
Up to last year, apart from the obvious examples of the 2 − (qn, q, 1) designs associated with
the affine geometry AG(n, q), all known strictly additive 2-designs had a quite “big" λ. Very
recently, a strictly additive 2 − (81, 6, 2) design has been found in [3]. This design, besides
being simple (the only design with these parameters previously known [2] has sixteen pairs of
repeated blocks), has the property that every block is union of two parallel lines of AG(4, 3).
In the attempt of getting other strictly additive designs with this property we found some
infinite series of 2-designs whose parameter-sets are probably new.
In this talk, besides presenting the above series, I will try to outline a proof that for every
odd k, there are infinitely many values of v for which a strictly additive 2 − (v, k, 1) design
exists.
References
[1] A. Caggegi, G. Falcone, M. Pavone, On the additivity of block designs, J. Algebr. Comb.
45, 271–294 (2017).
[2] H. Hanani, Balanced incomplete block designs and related designs, Discrete Math. 11
(1975), 255–369.
[3] A. Nakic, The first example of a simple 2 − (81, 6, 2) design, Examples and Counterexam-
ples 1 (2021).
263
and (r, λ) = 1, admitting a flag-transitive automorphism group G, mostly focusing on the
constructions of the various examples contained in it.
Testing isomorphism of circulant objects in polynomial time
Mikhael Muzychuk, misha.muzychuk@gmail.com
Ben-Gurion University of the Negev, Israel
Coauthor: Ilia Ponomarenko
We show that isomorphism testing of two cyclic combinatorial objects may be done in a poly-
nomial time of their sizes provided that both objects share the same regular cyclic group of
automorphisms given in advance.
Strictly additive 2-designs
Anamari Nakic´, anamari.nakic@fer.hr
University of Zagreb, Croatia
Coauthor: Marco Buratti
This work draws inspiration from an interesting theory developed in [1]. A design (V, B) is said
to be additive if V is a subset of an abelian group G and the elements of any block B ∈ B sum
up to zero. We propose to speak of a strictly additive design when V coincides with G.
Up to last year, apart from the obvious examples of the 2 − (qn, q, 1) designs associated with
the affine geometry AG(n, q), all known strictly additive 2-designs had a quite “big" λ. Very
recently, a strictly additive 2 − (81, 6, 2) design has been found in [3]. This design, besides
being simple (the only design with these parameters previously known [2] has sixteen pairs of
repeated blocks), has the property that every block is union of two parallel lines of AG(4, 3).
In the attempt of getting other strictly additive designs with this property we found some
infinite series of 2-designs whose parameter-sets are probably new.
In this talk, besides presenting the above series, I will try to outline a proof that for every
odd k, there are infinitely many values of v for which a strictly additive 2 − (v, k, 1) design
exists.
References
[1] A. Caggegi, G. Falcone, M. Pavone, On the additivity of block designs, J. Algebr. Comb.
45, 271–294 (2017).
[2] H. Hanani, Balanced incomplete block designs and related designs, Discrete Math. 11
(1975), 255–369.
[3] A. Nakic, The first example of a simple 2 − (81, 6, 2) design, Examples and Counterexam-
ples 1 (2021).
263
