Cyclic relative difference families with block size four and their applications

Given a subgroup H of a group $(G,+)$, a $(G,H,k,1)$ difference family (DF) is a set $F$ of k-subsets of G such that $\{f-f^{\prime }:f,f^{\prime }\in F,f\ne f^{\prime },F\in F\}=G\setminus H$. Let $g{Z}_{gh}$ be the subgroup of order h in $Z_{gh}$ generated by g. A $(Z_{gh},g{Z}_{gh},k,1)$-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 $gh\equiv h\left(\mathrm{mod}12\right)$, $g⩾4$ and $(g,h)\notin \left\{\right(9,3),(5,6\left)\right\}$. As a corollary, it is shown that a 1-rotational Steiner system S$(2,4,v)$ exists if and only if $v\equiv 4\left(\mathrm{mod}12\right)$ 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 $v\equiv 1,2,3,4,6\left(\mathrm{mod}12\right)$ 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.