Cyclic Designs with Block Size 4 and Related Optimal Optical Orthogonal Codes

Abstract

We prove the existence of a cyclic (4p, 4, 1)-BIBD—and hence, equivalently, that of a cyclic (4, 1)-GDD of type 4p—for any prime p \equiv 1 \pmod 6 such that (p−1)/6 has a prime factor q not greater than 19. This was known only for qe2, i.e., for p \equiv 1 \pmod 12. In this case an explicit construction was given for p \equiv 13 \pmod 24. Here, such an explicit construction is also realized for p \equiv 1 \pmod 24. We also give a strong indication about the existence of a cyclic (4p 4, 1)-BIBD for any prime p \equiv 1 \pmod 6, p>7. The existence is guaranteed for p>(2q3−3q2+1)2+3q2 where q is the least prime factor of (p−1)/6. Finally, we prove, giving explicit constructions, the existence of a cyclic (4, 1)-GDD of type 6p for any prime p>5 and the existence of a cyclic (4, 1)-GDD of type 8p for any prime p \equiv 1 \pmod 6. The result on GDD's with group size 6 was already known but our proof is new and very easy. All the above results may be translated in terms of optimal optical orthogonal codes of weight four with λe1.