On silver and golden optical orthogonal codes
Keywords:Silver and golden ratio, Pell and Fibonacci numbers, difference packing, optimal optical orthogonal code, strong difference family, difference multiset
It is several years that no theoretical construction for optimal (v, k, 1) optical orthogonal codes (OOCs) with new parameters has been discovered. In particular, the literature almost completely lacks optimal (v, k, 1)-OOCs with k > 3 which are not regular. In this paper we will show how some elementary difference multisets allow to obtain three new classes of optimal but not regular (3p, 4, 1)-, (5p, 5, 1)-, and (2p, 4, 1)-OOCs which are describable in terms of Pell and Fibonacci numbers. The OOCs of the first two classes (resp. third class) will be called silver (resp. golden) since they exist provided that the square of a silver element (resp. golden element) of ℤp is a primitive square of ℤp.