Classification of Maximal Optical Orthogonal Codes of Weight 3 and Small Lengths

Abstract

Dedicated to the memory of the late professor Stefan Dodunekov
on the occasion of his 70th anniversary.

We classify up to multiplier equivalence maximal (v, 3, 1) optical
orthogonal codes (OOCs) with v ≤ 61 and maximal (v, 3, 2, 1)
OOCs with v ≤ 99.

There is a one-to-one correspondence between maximal (v, 3, 1) OOCs,
maximal cyclic binary constant weight codes of weight 3 and minimum dis
tance 4, (v, 3; ⌊(v − 1)/6⌋) difference packings, and maximal (v, 3, 1) binary
cyclically permutable constant weight codes. Therefore the classification of
(v, 3, 1) OOCs holds for them too. Some of the classified (v, 3, 1) OOCs are
perfect and they are equivalent to cyclic Steiner triple systems of order v
and (v, 3, 1) cyclic difference families.