Anita Pasotti

Further progress on difference families with block size 4 or 5

A strong indication about the existence of a (7p, 4, 1) difference family with p ≡ 7 (mod 12) a prime has been given in [11]. 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.

New results on optimal (v, 4, 2, 1) optical orthogonal codes

We investigate further the existence question regarding optimal (v, 4, 2, 1) optical orthogonal codes begun in Momihara and Buratti (IEEE Trans Inform Theory 55:514–523, 2009). We give some non-existence results for infinitely many values of v ≡ ± 3 (mod 9) and several explicit constructions for infinite classes of perfect optical orthogonal codes.

On optimal (v, 5, 2, 1) optical orthogonal codes

The size of a (v, 5, 2, 1) optical orthogonal code (OOC) is shown to be at most equal to

A new result on the problem of Buratti, Horak and Rosa

{\lceil{\frac{v}{12}}\rceil}

A Generalization of the Problem of Mariusz Meszka

when v ≡ 11 (mod 132) or v ≡ 154 (mod 924), and at most equal to

α-Labelings of a Class of Generalized Petersen Graphs

{\lfloor{\frac{v}{12}}\rfloor}