Further progress on difference families with block size 4 or 5

A strong indication about the existence of a (7 p, 4, 1) difference family with p ≡ 7 (mod 12) a prime has been given in [1]. Here, developing some ideas of that paper, we give, much more generally, a strong indication about the existence of a cyclic (pq,4,1) difference family whenever p and q are primes congruent to 7 (mod 12) and of a cyclic (pq,5,1) difference family whenever p and q are primes congruent to 11 (mod 20). Indeed we give an algorithm for their construction that seems to be always successful and we have checked it works whenever both primes p and q do not exceed 1,000. All our (pq,4,1) and (pq,5,1) difference families have the nice property of admitting a multiplier of order 3 or 5, respectively, that fixes almost all base blocks. As an intermediate result we also find an optimal (p, 5, 1) optical orthogonal code for every prime p ≡ 11 (mod 20) not exceeding 10,000.