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 give some nonexistence results for GSEDFs. In particular, we prove that a $(v, 3;k_1,k_2,k_3; \lambda _1,\lambda _2,\lambda _3)$-GSEDF does not exist when $k_1+k_2+k_3< v$. We also give a first recursive construction for GSEDFs and prove that if there is a $(v,2;2\lambda ,\frac{v-1}{2};\lambda ,\lambda )$-GSEDF, then there is a $(vt,2;4\lambda ,\frac{vt-1}{2};2\lambda ,2\lambda )$-GSEDF with $v>1$, $t>1$ and $v\equiv t\equiv 1\pmod 2$. Then we use it to obtain some new GSEDFs for $m=2$. In particular, for any prime power q with $q\equiv 1\pmod 4$, we show that there exists a $(qt, 2;(q-1)2^{n-1},\frac{qt-1}{2};(q-1)2^{n-2},(q-1)2^{n-2})$-GSEDF, where $t=p_1p_2\dots p_n$, $p_i>1$, $1\le i\le n$, $p_1, p_2,\dots ,p_n$ are odd integers.