# Two-Dimensional Permutation Vectors' (PV) Code for Optical Code Division Multiple Access Systems
## 1 Introduction  
Driven by the ever-growing data size from internet usage rate, optical code division multiple access (OCDMA) system became the center of attention due to its ability to overcome the bottleneck problem for data-hungry applications with massive data rate. OCDMA systems permit multiple subscribers to asynchronously and concurrently access the medium without any contention along with the ability to provide reliable bandwidth at relatively low cost [1, 2]. Spectral amplitude coding (SAC) technique is measured as a proficient scheme in alleviating the effects of multiple access interference (MAI) and its inherited intensity noise [3]. Along with its MAI alleviation features, SAC-OCDMA system is implemented with low cost and less complexity with broadband incoherent sources such as light emitting diodes (LED) [4, 5]. Two dimensional (2D) coding schemes are enthusiastically adapted to increase orthogonality which lead to efficient bandwidth utilization. Further, performance is improved in terms of increased subscribers due to spectral density expansion at the cost of high-speed electronic devices [6]. By combining two different dimensions of CDMA, two-dimensional (2D) codes have been proposed for OCDMA networks. SAC-based systems mostly use spectral/spatial (S-S) and wavelength/time (W-T)-coding approaches for 2D coding [7; 8]. In the case of 2D (W-T) scheme, a different wavelength is assigned to each chip and placed across the bit time period, which further offers higher transmission capacity and more flexibility when compared with 1D coding schemes [9; 10; 11]. Further, to improve bit error rate (BER) and cardinality for a certain data rate, AND detector circuit is used at the receiver [12]. AND detector circuit composed of parallel structure of two photo detectors or photodiodes (PD and s-PD) which are connected electrically in opposition and power difference of the two optical inputs is the resultant output signal. The 2D W-T optical orthogonal codes (OOC) with constant weight and variable code length properties were discussed by [PERSON] et al. [13]. The 2D OOC code provides multimedia services with district service requirements for optical CDMA networks by designing encoder and decoders with fiber bragg gratings (FBGs). A new family of 2D hybrid W-T code was proposed by [PERSON] et al. [14], by combining OCC and balanced incomplete block design zero cross-correlation (BIBD-ZCC) codes. The 2D hybrid code provides both off-peak auto-correlation and cross-correlation features with time-spreading and wavelength-hopping patterns. The presented code is suitable for the asynchronous and synchronous OCDMA environment with benefits such as high cardinality and good correlation properties. The 2D W-T OCDMA system, which is based on coherent light sources, was analyzed in the presence of beat noise (BN) and MAI effects by [PERSON] et al. [15]. For mitigating BN, a number of aspects for 2D Time/Wavelength (TW) code designs are presented along with BN dependence on code properties. Various 2D W-T codes based on single pulse per column/row and multiple pulse per column/row were compared. A new technique for designing OCDMA scheme with time spread and wavelength group hopping embedded maximum length (ML or m) sequence code was presented by [PERSON] et al. [16]. This technique uses cyclic and periodic free-spectral-range properties of arrayed waveguide gratings (AWGs) routers to design 2D time-spreading and wavelength group-hopping embedded m-sequence code. The code showed cross-correlation ((\(\lambda\)c) = G \(\times\) w), where G stands for number of wavelengths and \(w\) referred as code weight. Fine AWG produces m-sequence code words, which is further time spread in the wavelength domain using a number of coarse AWGs. Further, to design 2D zero cross-correlation (ZCC) code, W-T coding approach was used to obtain an increased number of subscribers by [PERSON] et al. [17]. In this work, 2D ZCC codes were designed based on 1D ZCC codes while conserving same code length and correlation constraint.