Some progress on (v,4,1) difference families and optical orthogonal codes

Abstract

Some new classes of optimal (v,4,1) optical orthogonal codes are constructed. First, mainly by using perfect difference families, we establish that such an optimal OOC exists for v⩽408, v≠25. We then look at larger (p,4,1) OOCs with p prime; some of these codes have the nice property that the missing differences are the (r−1)th roots of unity in Z_p (r being the remainder of the Euclidean division of p by 12) and we prove that for r=5 or 7 they give rise to (rp,4,1) difference families. In this way we are able to give a strong indication about the existence of (5p,4,1) and (7p,4,1) difference families with p a $\text{prime}\text{≡5,7}\text{mod}\text{12}$ respectively. In particular, we prove that for a given prime $\text{p≡7}\text{mod}\text{12}$, the existence of a (7p,4,1) difference family is assured (1) if p<10,000 or (2) if ω is a given primitive root unity in Z_p and we have $\text{3≡ω}{}^{\mathrm{i}}\text{(}\text{mod}\text{p)}$ with $\text{gcd}\text{(i,}\text{p−1}\text{6}\text{)<20}$.

Finally, we remove all undecided values of v⩽601 for which a cyclic (v,4,1) difference family exists, and we give a few cyclic pairwise balanced designs with minimum block size 4.