Cyclic relative difference families with block size four and their applications

Abstract

Given a subgroup $H$ of a group $(G,+)$, a $(G,H,k,1)$ difference family (DF) is a set $\mathcal{F}$ of $k$-subsets of $G$ such that $\{f-f^{\prime }:f,f^{\prime }\in F,f\ne f^{\prime },F\in \mathcal{F}\}=G\setminus H$. Let $g{\mathbb{Z}}_{gh}$ is the subgroup of order $h$ in ${\mathbb{Z}}_{gh}$ generated by $g$. A $({\mathbb{Z}}_{gh},g{\mathbb{Z}}_{gh},k,1)$-DF is called cyclic and written as a $(gh,h,k,1)$-CDF. This paper shows that for $h\in \{2,3,6\}$, there exists a $(gh,h,4,1)$-CDF if and only if $gh\equiv h(\mathrm{mod}12)$, $g\ge 4$ and $(g,h)\notin \{(9,3),(5,6)\}$. As a corollary, it is shown that a 1-rotational S$(2,4,v)$ exists if and only if $v\equiv 4(\mathrm{mod}12)$ and $v\ne 28$. This solves the long-standing open problem on the existence of a 1-rotational S$(2,4,v)$. As another corollary, we establish the existence of an optimal $(v,4,1)$-optical orthogonal code with $\lfloor (v-1)/12\rfloor$ codewords for any positive integer $v\equiv 1,2,3,4,6(\mathrm{mod}12)$ and $v\ne 25$. We also give applications of our results to cyclic group divisible designs with block size four and optimal cyclic $3$-ary constant-weight codes with weight four and minimum distance six.