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

Abstract

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

“…Note, in particular, that a m-regular (v, k, 1)-OOC is a (v, m, k, 1) relative difference family [9] As already remarked in [11], a codeword-set C of a (v, k, λ a , 1)-OOC is of odd type if and only if v is even and v 2 ∈ C. Thus a (v, k, λ a , 1)-OOC may have at most one codeword-set of odd type otherwise (2 ) would be contradicted.…”

“…The present paper deals with (v, 5, 2, 1)-OOCs and it is the natural continuation of [11] and [25]. As one could expect, many results are, in a certain sense, "twin results" of those papers.…”

“…For instance, it was proved in [11] and [25] that a tight upper bound on the size of a (v, 4, 2, 1)-OOC is the floor or the ceiling of v 8 according to the congruence class of v. The twin result here is that a tight upper bound on the size of a (v, 5, 2, 1)-OOC is the floor or the ceiling of v 12 . On the other hand some constructions in this paper are considerably more difficult than their "twins" in [11] and [25]; in various cases we need to invent "ad hoc" tricks to obtain our codes. For instance, we mention the use of some perfect or near-perfect matching of a suitable hypergraph described in the third section to try to obtain an optimal (v, 5, 2, 1)-OOC with v a prime ≡ 1 or 5 (mod 12).…”

“…Several results about (v, k, k − 1, 1)-OOCs (the so called conflict-avoiding codes) have been recently obtained in [23,24] while (v, 4, 2, 1)-OOCs have been deeply investigated in [5,11,25,26].…”

A clique-based online algorithm for constructing optical orthogonal codes

“…But, it is still difficult to find a complete solution for larger values of k, λ a , and λ c for the moment. For example, only some partial answers have been achieved for the cases of k = 4 or 5, λ a ≤ 2 and λ c ≤ 2 [2,3,12,13,14,18]. Further details can be found in the references and therein.…”