# Two-Dimensional Permutation Vectors' (PV) Code for Optical Code Division Multiple Access Systems
## 1 Introduction
The proposed system architecture is simple and cost effective due to direct detection technique employed at the receiver. Multi-diagonal prime hop code (2D MDPHC) based on W-T coding technique and with zero cross-correlation properties was presented by [PERSON] et al. [18]. The presented code is a combination of two different 1D codes (MD and prime code) and can accommodate a larger number of users with an increased data transmission rate. The entropy in information theory explains the disorder of information. These types of imports are important and necessary because a well-developed identity in one discipline will bring definition to a phenomenon as well as mathematical tools and theory. Our aim here was to obtain crisp results in a simple setting which can be used to understand the basic trade-offs between the intrinsic entropy rate of the system and the available rate on the optical communication [19]. A passive optical network (PON) system based on 2D-OCDMA wavelength division multiplexing (WDM) was presented by [PERSON] et al. [20] in which a hybrid of 2D prime hop system (PHS) codes and 2D hybrid coded (HC) signature codes were employed for uplink and WDM scheme for the downlink. A vertical cavity surface emitting laser was deployed with an avalanche photo detector receiver and optical line terminal unit in the optical network. The system performance was analyzed for the BER, signal to noise (SNR), optical power budget, and quality (Q) factor. BER improvement (\(10^{-10}\) at data rates 40 Giga bits per second (Gbps)) as well as long PON distance up to 41 km with increased maximum throughput equal to 285.1 Gb/s/km was achieved. Moreover, [PERSON] et al. [21] proposed the construction of 2D diagonal eigenvalue unity (DEU) code using a spectral-spatial (S-S) coding approach with minimum cross-correlation. Results indicated that the 2D DEU provides an increase in cardinality when compared to existing 2D diluted perfect difference DPD and 1D DEU codes. [PERSON] et al. [22] presented an analytical model consisting of a hybrid of all-optical orthogonal frequency division multiplexing (AO-OFDM) and OCDMA. The presented model incorporated probabilistic subcarrier overlapping and MAI testing capabilities with over amplifier-free long-reach passive optical networks (LR-PONs) using cost-effective intensity modulations and direct detection (IM-DD) techniques. The proposed system considered subcarrier hopping by utilizing 2D-HC codes. It was observed that the hybrid (AO-OFDM) and OCDMA with 2D-HC outperformed a traditional multichannel OCDMA system for any number of simultaneous users and low received powers in comparison with 1D-Walsh Hadamard, 1D, and 2D prime codes. Numerical results showed that 16-quadrature amplitude modulation (16-QAM) AO-OFDM-OCDMA provides comparable performance to traditional multichannel 16-QAM coherent optical OFDM in the downstream direction. The presented technique supported up to 58 km with a maximum 45 users, without complex coherent technology. In addition, at 108 km as a maximum reachable distance (at 40 Gb/s), QAM signal was achieved in budget power calculation while considering standard forward error correction (FEC) techniques. Further, a recent study conducted by [PERSON] et al. [23] presented performance analysis of an OCDMA system for LR-PON systems, considering MAI, receiver noise, and single-mode fiber (SMF) channel effects. A mathematical model representing 2D optical code parameters for different receiver structures using Matlab simulations was developed including effects of channel imperfections, such as attenuation losses and chromatic dispersion. Probability of error was investigated for back-to-back (B2B) single mode fiber (SMF) with conventional correlation receiver (CCR) and SMF channel with successive interference cancelation (SIC) receiver. Performance improvement, in terms of number of simultaneous users with Q factor (6) at fiber distances 190 and 120 km, was achieved without amplification. The rest of this manuscript is arranged as follows.