Abstract:
 1. A coverfree family (introduced by Kautz and Singleton (1964)) is a family of subsets of a finite set in which no one is covered by the union of $r$ others. We study a variation of coverfree family: A binary matrix is ${(r, w]consecutivedisjunct}$ if for any $w$
cyclically consecutive columns $C_1,\cdots, C_w$ and another $r$ cyclically consecutive columns $C_{w+1},\cdots, C_{w+r}$, there exists one row intersecting $C_1, \cdots C_w$ but none of $C_{w+1},\cdots, C_{w+r}$. In this talk, I will present how we apply the algorithmic Lovasz Local Lemma to obtain an upper bound of the minimum number of rows in an $(r, w]$consecutivedisjunct matrix of $n$ columns. In group testing, our goal is to determine a small subset of positive items in a large population by group tests. In this talk, I will also introduce how we exploit consecutivedisjunct matrices to solve threshold group testing of consecutive positives (introduced by Chang and Tsai (2013)). This is a joint work with YiChang Chiu and YiLin Tsai.
2. Let $\Gamma$ denote a distanceregular graph with diameter $D \geq 3$ and intersection numbers $a_1=0, a_2 \neq 0$, and $c_2=1$. We show the connection between the $d$bounded property and the nonexistence of parallelograms of any length up to $d+1$. Assume further that $\Gamma$ is with classical parameters $(D, b, \alpha, \beta)$, Pan and Weng $(2009)$ showed that $(b, \alpha, \beta)= (2, 2, ((2)^{D+1}1)/3).$ Under the assumption $D \geq 4$, we exclude this class of graphs by an application of the above connection.
This is a joint work with Yehjong Pan and Chihwen Weng.
