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

TL;DR

This work addresses constructing large optical orthogonal codes (OOCs) with favorable autocorrelation and cross-correlation properties by leveraging multi-orbit cyclic subspace codes and multi-Sidon spaces in extension fields. The authors develop a multiplicative-structure approach that translates OOC design into affine-subspace configurations via a $\theta$ map between binary vectors and subsets of $\mathbb{Z}_n$, enabling OOC construction from cyclic subspace codes. They prove that unions of distinct orbits in cyclic subspace codes yield large, new OOC families when mapped back, and they provide explicit constructions using linearized polynomials that give new parameter sets, including $(q^{2k}-1, q^k, q)$ with sizable code sizes. These results expand the repertoire of OOCs beyond existing optimal and asymptotically optimal families, with potential impact on OCDMA systems. All formulas are expressed with $...$ notation where appropriate to ensure precise mathematical characterization.

Abstract

We present a new application of multi-orbit cyclic subspace codes to construct large optical orthogonal codes, with the aid of the multiplicative structure of finite fields extensions. This approach is different from earlier approaches using combinatorial and additive (character sum) structures of finite fields. Consequently, we immediately obtain new classes of optical orthogonal codes with different parameters.

Theorems & Definitions (19)

• Definition 1.1
• Theorem 1.2: see johnson1962new
• Definition 2.1
• Lemma 2.2
• Theorem 2.3
• proof
• Theorem 3.1: see roth2017construction
• Proposition 3.2
• proof
• Theorem 3.3