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 $v \leq 408$; $v \neq 25$. 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 $\equiv 5, 7$ mod 12 respectively. In particular, we prove that for a given prime $p\equiv 7$ 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 $3 \equiv \omega^i$ mod p with $\gcd(i; {p-1\over 6})<20$. Finally, we remove all undecided values of v < 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.
Some progress on (v,4,1) difference families and optimal optical orthogonal codes
BURATTI, Marco
2004
Abstract
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 $v \leq 408$; $v \neq 25$. 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 $\equiv 5, 7$ mod 12 respectively. In particular, we prove that for a given prime $p\equiv 7$ 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 $3 \equiv \omega^i$ mod p with $\gcd(i; {p-1\over 6})<20$. Finally, we remove all undecided values of v < 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
