Mathematical Research Seminar - Archive

Let γ be a Cayley graph over a dihedral group D_2n (a dihedrant for short) and G the group of automorphisms of γ. Suppose G acts transitively on the edges of γ. The problem of characterizing such graphs was proposed by Song et al. It is currently solved only under additional assumptions on γ  or G.

In this talk, we introduce two new infinite families of edge-transitive dihedrants and show that the graph γ is either described in the earlier papers, belongs to one of the two new families, or the group G satisfies certain conditions. Using these conditions, we also classify γ in the case when G is a solvable group. This generalizes a result of Pan et al. dealing with the case where (D_2n)_R is normal in G.

This is a joint work with István Kovács.

A tight point set T  in a finite classical polar space is a collection of points with as many pairs of collinear points as possible. To be more precise, the point-collinearity graph of a finite classical polar space P is strongly regular, and the eigenvalues of such a graph give a bound on the number of adjacent vertices in T  in terms of |T|; when this bound is met exactly, we say that T  is a tight set of P . The existence and classification of tight sets gives interesting information about the structure of the polar spaces, and they can also be used to define new strongly regular graphs.

We will specifically be looking at examples of tight sets in the hyperbolic polar spaces Q ⁺(2d-1,q). The tight sets of $ Q ⁺(5,q) have been well studied, since under the Klein correspondence they are equivalent to the study of Cameron-Liebler line classes of PG(3,q). However up to this point, there are no known nontrivial examples of tight sets in Q ⁺(7,q).

In this talk, we describe a method to construct new nontrivial tight sets in Q ⁺(7,q) based on the embedding of a Segre variety S_1,3 in the hyperbolic space, along with some resulting examples. We further look at how this construction can be extended to Q ⁺(2d-1,q) for larger even values of d.

Joint work with Jozefien D'haesseleer.

Extremal Type II ℤ₂^m-codes are a class of self-dual ℤ₂^m -codes with Euclidean weights divisible by 2m+1 and the largest possible minimum Euclidean weight for a given length.

We introduce a doubling method for constructing a Type II ℤ₂^m -code of length n from a known Type II ℤ₂^m -code of length n. Based on this method, we develop an algorithm to construct non-free extremal Type II ℤ₂^m -codes starting from a free extremal Type II ℤ₂^m -code with an extremal ℤ₂^m-1 -residue code and length 24, 32 or 40.

We obtain at least 10 new extremal Type II ℤ₈ -codes of length 32 and at least 11 new extremal Type II ℤ₁₆ -codes of length 32.

Joint work with Sanja Rukavina.