2015 Workshop on Combinatorics and Applications at SJTU

April 21 -- 27, Shanghai Jiao Tong University

Date: Tuesday, April 21, 2015

Time: 11:45 - 12:15

Venue: Large Meeting Room, Math Building

Title: Tight Relative $2e$-Designs in Association Schemes

Abstract:

Relative $t$-designs is defined on Q-polynomial association scheme and we call it tight if it satisfies the Fisher type lower bound. We will mainly review some results about tight relative $2$-designs $(Y,w)$ on two shells $X_{r_1} \cup X_{r_2}$ in binary Hamming association scheme $H(n,2)$ and Johnson association scheme $J(v,k)$. The good feature of $H(n,2)$ is that the distance set $\{ \alpha_1,\alpha_2,\gamma \}$ and $\frac{w_2}{w_1}$ are uniquely expressed in terms of $n,r_1,r_2,N_{r_1}$. This implies coherent configuration is attached to $Y$. (However, for $J(v,k)$, this property is difficult to prove in general .) There exist many tight relative $2$-designs in $H(n,2)$ with both constant weight and $w_2 \neq w_1$. So far, we are unable to find such designs with $w_2 \neq w_1$ in $J(v,k)$. Now we are working on the existence of tight relative $4$-designs on two shells in $H(n,2)$. In this case, we can not determine all the feasible parameters. But it is proved that $Y \cap X_{r_i}, i=1,2$ should be combinatorial $3$-design. Finally this problem is related to the existence of some combinatorial designs. This is joint work with Eiichi Bannai and Etsuko Bannai.

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