Cyclic designs with block size 4 and related optimal optical orthogonal codes

We prove the existence of a cyclic (4 p, 4, 1)-BIBD—and hence, equivalently, that of a cyclic (4, 1)-GDD of type $4^p$—for any prime p ≡ 1 (mod 6) such that (p−1)/6 has a prime factor q not greater than 19. This was known only for q = 2, i.e., for p ≡ 1 (mod 12). In this case an explicit construction was given for p ≡ 13 (mod 24). Here, such an explicit construction is also realized for p ≡ 1 (mod 24). We also give a strong indication about the existence of a cyclic (4 p, 4, 1)-BIBD for any prime p ≡ 1 (mod 6), p > 7. The existence is guaranteed for $p > (2q^3 − 3q^2 + 1)^2 + 3q^2$ 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 $6^p$ for any prime p > 5 and the existence of a cyclic (4, 1)-GDD of type $8^p$ for any prime p ≡ 1 (mod 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 λ = 1.