# KPPY 82

Hirotake Kurihara, Alexander Gavrilyuk, Takayuki Okuda, Minki Kim, Jaeseon Kim

April 22, 2017. 11:00am-5:30pm at YNU

Schedule | ||

11:00am - 11:50 |
Hirotake Kurihara Kitakyushu College (Japan) | Euclidean distortions of distance regular graphs |

12pm - 1:25 | Lunch | |

1:30 - 2:20 |
Alexander Gavrilyuk USTC (China) | On tight sets of hyperbolic quadrics |

2:30 - 3:20 |
Takayuki Okuda Hiroshima University (Japan) | Distance transitive graphs coming from great antipodal sets on symmetric R-spaces |

3:30 - 4:20 |
Minki Kim KAIST (South Korea) | Nerves and minors |

4:30 - 5:20 |
Jaeseon Kim POSTECH (South Korea) | Construction of low-weight linear codes from defining sets |

## Abstracts

Hirotake Kurihara

Euclidean distortions of distance regular graphs

Euclidean distortions of distance regular graphs

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.

Alexander Gavrilyuk

On tight sets of hyperbolic quadrics

On tight sets of hyperbolic quadrics

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.

Takayuki Okuda

Distance transitive graphs coming from great antipodal sets on symmetric R-spaces

Distance transitive graphs coming from great antipodal sets on symmetric R-spaces

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).

Minki Kim

Nerves and minors

Nerves and minors

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.

Jaeseon Kim

Construction of low-weight linear codes from defining sets

Construction of low-weight linear codes from defining sets

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.