KPPY 82

Let $X$ be a metric space and $F$ be an embedding from $X$ to the $\ell_2$-Hilbert space. The distortion of $F$ is defined by the product of the Lipschitz constant of $F$ and the Lipschitz constant of $F^{-1}$, and the Euclidean distortion of $X$ is defined by the infimum of distortion amongst the embedding of $X$. It is not easy to determine the Euclidean distortion of a given metric space. In this talk, we will discuss the Euclidean distortions of distance-regular graphs. In particular, we will give shape bounds of the Euclidean distortions of distance-regular graphs, and when the diameter is small, we will give the explicit values of the Euclidean distortions.

A set of points $M$ of a finite polar space $\mathcal{P}$ is called tight, if the average number of points of $M$ collinear with a given point of $\mathcal{P}$ equals the maximum possible value. In the case when $\mathcal{P}$ is a hyperbolic quadric $Q^+(2n+1,q)$, the notion of tight sets generalises that of Cameron-Liebler line classes in $PG(3,q)$, whose images under the Klein correspondence are the tight sets of the Klein quadric $Q^+(5,q)$. Very recently, some new constructions and necessary conditions for the existence of Cameron-Liebler line classes have been obtained. In this talk, we will discuss a possible extension of these results to the general case of tight set of hyperbolic quadrics.

The concept of great antipodal sets on symmetric spaces was introduced by Chen--Nagano [Trans. Amer. Math. Soc. (1988)]. For each symmetric R-space, great antipodal sets on it are known to be unique up to isometries. In this talk, we define a natural graph structure on each great antipodal set on symmetric R-spaces and determine all such graphs as distance transitive graphs. This is a joint work with Hirotake Kurihara (National Institute of Technology, Kitakyushu College).

Given a finite family of simplicial complexes $\mathcal{F}=\{K_1,\dots, K_n\}$, a typical nerve theorem asserts that if $\bigcap_{i\in\sigma} K_i$ is sufficiently connected whenever it is non-empty, then the nerve of $\mathcal{F}$ adequately reflects the topology of $\bigcup_{i\in[n]} K_i$. In this talk, we consider a finite family $\mathcal{F}$ of induced subgraphs of a simple undirected connected graph $G$. In particular, we will see how topology of the nerve of $\mathcal{F}$ reflects the forbidden minor conditions on $G$. This is joint work with Andreas Holmsen and Seunghun Lee.

An $[n, k, d]$-(binary) linear code $C$ is a subspace of $\mathbb{F}_2^n$ and dimension $k$ with the minimum (Hamming) distance $d$. We say that a linear code $C$ is $t$-weight if the number of non-zero weight is equal to $t+1$. A low weight code is interesting object in combinatorics. In particular, $1$-weight codes give a bound of constant-weight code in coding theory. $2$-weight codes are related to a strongly regular graph which is important object in graph theory. $3$-weight codes imply some three-class association scheme, etc. We consider the following linear code Let $D$ be a subset of $\mathbb{F}_2^n$. We define \[ \mathcal{C}_D = \{c(s,u)= (s+u\cdot x)_{x\in D^*} | s\in\mathbb{F}_2, u\in\mathbb{F}_2^n\}. \] Then $\mathcal{C}_D$ is a linear code of length $|D^*|$ and dimension at most $n+1$. We call $D$ the defining set of $ \mathcal{C}_D$. In this talk, we explicitly determine the weight distribution of this code when $D$ is a linear code or the support of a bent function. And, we consider subcodes of $\mathcal{C}_D$ to obtain a low-weight codes. Finally, we will introduce new operation about these codes and get more low-weight codes.