# 3-Dimensional Optical Orthogonal Codes with Ideal Autocorrelation-Bounds and Optimal Constructions

## TL;DR

This paper introduces new 3D optical orthogonal codes with ideal autocorrelation and near-optimal size, constructed using automorphisms of finite projective and affine geometries, demonstrating their optimality and tight bounds.

## Contribution

It presents novel constructions of 3D optical orthogonal codes with ideal autocorrelation and optimal size, utilizing automorphisms of finite geometries, and establishes tight bounds for these codes.

## Key findings

- Codes have ideal autocorrelation ($$) and cross correlation ($1$).
- All codes are optimal with respect to the Johnson bound.
- Constructed using automorphisms of finite projective and affine geometries.

## Abstract

Several new constructions of 3-dimensional optical orthogonal codes are presented here. In each case the codes have ideal autocorrelation $\mathbf{ \lambda_a=0} $, and in all but one case a cross correlation of $ \mathbf{\lambda_c=1} $. All codes produced are optimal with respect to the applicable Johnson bound either presented or developed here. Thus, on one hand the codes are as large as possible, and on the other, the bound(s) are shown to be tight. All codes are constructed by using a particular automorphism (a Singer cycle) of $ \mathbf{ PG(k,q)} $, the finite projective geometry of dimension $ k $ over the field of order $ \mathbf{q} $, or by using an affine analogue in $ AG(k,q) $.