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

摘要

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 less than or equal to 408, v not equal 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 prime equivalent to 5, 7 mod 12 respectively. In particular, we prove that for a given prime p = 7 mod 12, the existence of a (7p, 4, 1) difference family is assured (1) if p < 10,000 or (2) if omega is a given primitive root unity in Z(p), and we have 3 equivalent to omega' (mod p) with gcd(i, p-1/6) < 20.Finally, we remove all undecided values of v less than or equal to 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. (C) 2004 Elsevier Inc. All rights reserved.