Tayuan Huang

The notion of $d$-disjunct matrices was first introduced by Kautz and Singleton in 1964, and has been generalized to$(s,l)$-superimposed codes and designs by D'yachkov and Torney $et. al.$ in 2002, and to $(s,l;e)$- generalized cover free families recently by Stinson. All these structures can be used in combinatorial group testing algorithms applicable to DNA library screening, and they are therefore called pooling designs. As pointed out by D.-Z. Du that the structure of association schemes may play an important role in improving the performance of pooling designs, a few families of $(s,l,e)$-disjunct matrices defined over Johnson graphs and Grassmann graphs were previously considered. Following this way, error - correcting pooling designs based on incidence matrices associated with distance regular graphs over some finite geometries will be considered in this talk, including the symplectic geometry and the attenuated spaces. It then leads to the notion of pooling spaces, i.e., ranked partially ordered sets with certain atomic intervals. Optimal error correcting capability for a few families of pooling designs derived from pooling spaces with intervals carrying the structure of projective geometries will be considered. A decoding algorithm for pooling designs defined over $(s,l,e)$- disjunct matrices will also be included in this talk. Moreover, we will show that $(s,l,e)$- disjunct matrices also provide a class of pooling designs over complexes, called setwise group testing, which fits the practical consideration better.