Optical orthogonal codes and arcs in PG(d,q)

Abstract

We present a new construction for (n, w, λ)-optical orthogonal codes (OOCs). The construction is pleasingly simple, where codewords correspond to arcs, specifically normal rational curves. Moreover, our construction yields for each λ > 1 an infinite family of OOCs which are asymptotically optimal (with respect to the Johnson bound).

Using multi-orbit cyclic subspace codes for constructing optical orthogonal codes

“…For the parameters of known asymptotically optimal OOCs with λ = 2, please consult Table 2 of [7]. By using normal rational curves in PG(d, q), in [2], the authors constructed a family of asymptotically optimal ( q λ+2 −1 q−1 , q + 1, λ) OOCs, for any choice of λ > 1 and q > λ. In [4], by using normal rational curves and Singer groups, the authors constructed asymptotically optimal OOCs generalizing and improving previous constructions of OOCs, for instance those from conics [23] and arcs [2].…”

“…By using normal rational curves in PG(d, q), in [2], the authors constructed a family of asymptotically optimal ( q λ+2 −1 q−1 , q + 1, λ) OOCs, for any choice of λ > 1 and q > λ. In [4], by using normal rational curves and Singer groups, the authors constructed asymptotically optimal OOCs generalizing and improving previous constructions of OOCs, for instance those from conics [23] and arcs [2]. Among the various large constructions of OOCs, the authors have constructed a family of asymptotically optimal designs denoted as ( q h+1 −1 q−1 , q + 1, λ), where m and λ are integers satisfying h > λ ≥ 2, and q is a prime power such that q ≥ λ (refer to [4, Theorem 2]).…”