n-Dimensional Optical Orthogonal Codes, Bounds and Optimal Constructions

TL;DR

This paper extends optical orthogonal codes to higher dimensions, establishing bounds on their capacity and presenting new constructions for optimal and asymptotically optimal codes using finite projective spaces.

Contribution

It generalizes the concept of optical orthogonal codes to n-dimensions, derives bounds based on the Johnson bound, and introduces two novel constructions of ideal codes.

Findings

Established upper bounds on n-dimensional OOC capacity.

Presented an infinite family of optimal codes for each n ≥ 2.

Provided an asymptotically optimal code family for each n ≥ 2.

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 $PG(k,q)$.