Some results on generalized strong external difference families

Abstract

A generalized strong external difference family (briefly $(v,m;k_1,\dots ,k_m;{\lambda }_1,\dots ,{\lambda }_m)$-GSEDF) was introduced by Paterson and Stinson in 2016. In this paper, we construct some new GSEDFs for $m=2$ and use them to obtain some results on graph decomposition. We also give some nonexistence results for GSEDFs. Especially, we prove that a $(v,3;k_1,k_2,k_3;{\lambda }_1,{\lambda }_2,{\lambda }_3)$-GSEDF does not exist when $k_1+k_2+k_3<v$.

generalized strong external difference family; difference set; character theory; graph decomposition; nonexistence