N-dimensional optical orthogonal codes, bounds and optimal constructions

Abstract

We generalized to higher dimensions the notions of optical orthogonal codes. We establish uper bounds on the capacity of general $ n $-dimensional OOCs, and on specific types of ideal codes (codes with zero off-peak autocorrelation). The bounds are based on the Johnson bound, and subsume many of the bounds that are typically applied to codes of dimension three or less. We also present two new constructions of ideal codes; one furnishes an infinite family of optimal codes for each dimension $ n\ge 2 $, and another which provides an asymptotically optimal family for each dimension $ n\ge 2 $. The constructions presented are based on certain point-sets in finite projective spaces of dimension $k$ over $GF(q)$ denoted $PG(k,q)$.