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′:f,f′∈F,f≠f′,F∈F}=G∖H. Let gZgh be the subgroup of order h in Zgh generated by g. A (Zgh,gZgh,k,1)-DF is called cyclic and written as a (gh,h,k,1)-CDF. This paper shows that for h∈{2,3,6}, there exists a (gh,h,4,1)-CDF if and only if gh≡h(mod12), g⩾4 and (g,h)∉{(9,3),(5,6)}. As a corollary, it is shown that a 1-rotational Steiner system S(2,4,v) exists if and only if v≡4(mod12) and v≠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 ⌊(v−1)/12⌋ codewords for any positive integer v≡1,2,3,4,6(mod12) and v≠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.