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 (5 p,4,1) and (7 p,4,1) difference families with p a prime≡5,7 mod 12 respectively. In particular, we prove that for a given prime p≡7 mod 12 , the existence of a (7 p,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 3≡ω i ( mod p) with gcd(i, p−1 6 )<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.