New 22-designs from strong difference families

Strong difference families are an interesting class of discrete structures which can be used to derive relative difference families. Relative difference families are closely related to $2$-designs, and have applications in constructions for many significant codes, such as optical orthogonal codes and optical orthogonal signature pattern codes. In this paper, with a careful use of cyclotomic conditions attached to strong difference families, we improve the lower bound on the asymptotic existence results of $(\mathbb{F}_{p}\times \mathbb{F}_{q},\mathbb{F}_{p}\times \{0\},k,\lambda)$-DFs for $k\in\{p,p+1\}$. We improve Buratti's existence results for $2$-$(13q,13,\lambda)$ designs and $2$-$(17q,17,\lambda)$ designs, and establish the existence of seven new $2$-$(v,k,\lambda)$ designs for $(v,k,\lambda)\in\{(694,7,2),(1576,8,1),(2025,9,1),(765,9,2),(1845,9,2),(459,9,4)$, $(783,9,4)\}$.Comment: Version 1 is named "Improved cyclotomic conditions leading to new 2-designs: the use of strong difference families". Major revision according to the referees' comment

Enhanced 3-D OCDMA code family using asymmetric run length constraints

Abstract : This paper suggests an enhanced performance of the 3-D optical code division multiple access (OCDMA) codes, a space/wavelength/time spreading family of codes. The initial codes are in the format wavelength hopping/time sequence (WH/TS), selected according to their performance requirements and the TS sequence is constructed to achieve a linear space- time complexity. The asymmetric run length constraints are introduced in that regard, such that the positive bit positions align with the encoder/decoder frequency spacing pattern, yielding a 3-D WH/WS/TS. The selected 2-D OCDMA codes are one- coincidence frequency hopping codes (OCFHC) and optical orthogonal codes (OOC). As a time sequence code, the OOC code length is extended with a code rate of 0.04. The complexity and the bit error rate (BER) are herein given and compared with previous work. The results of the performance show not only an improvement in the number of simultaneous users due to the code length extension, but better correlation properties and hence a better signal-to-noise ratio

Deterministic Construction of Binary, Bipolar and Ternary Compressed Sensing Matrices

In this paper we establish the connection between the Orthogonal Optical Codes (OOC) and binary compressed sensing matrices. We also introduce deterministic bipolar $m\times n$ RIP fulfilling $\pm 1$ matrices of order $k$ such that $m\leq\mathcal{O}\big(k (\log_2 n)^{\frac{\log_2 k}{\ln \log_2 k}}\big)$. The columns of these matrices are binary BCH code vectors where the zeros are replaced by -1. Since the RIP is established by means of coherence, the simple greedy algorithms such as Matching Pursuit are able to recover the sparse solution from the noiseless samples. Due to the cyclic property of the BCH codes, we show that the FFT algorithm can be employed in the reconstruction methods to considerably reduce the computational complexity. In addition, we combine the binary and bipolar matrices to form ternary sensing matrices ($\{0,1,-1\}$ elements) that satisfy the RIP condition.Comment: The paper is accepted for publication in IEEE Transaction on Information Theor

Fast Decoder for Overloaded Uniquely Decodable Synchronous CDMA

We consider the problem of designing a fast decoder for antipodal uniquely decodable (errorless) code sets for overloaded synchronous code-division multiple access (CDMA) systems where the number of signals K_{max}^a is the largest known for the given code length L. The proposed decoder is designed in a such a way that the users can uniquely recover the information bits with a very simple decoder, which uses only a few comparisons. Compared to maximum-likelihood (ML) decoder, which has a high computational complexity for even moderate code length, the proposed decoder has a much lower computational complexity. Simulation results in terms of bit error rate (BER) demonstrate that the performance of the proposed decoder only has a 1-2 dB degradation at BER of 10^{-3} when compared to ML

A NOVEL CONSTRUCTION OF VECTOR COMBINATORIAL (VC) CODE FAMILIES AND DETECTION SCHEME FOR SAC OCDMA SYSTEMS

There has been growing interests in using optical code division multiple access (OCDMA) systems for the next generation high-speed optical fiber networks. The advantage of spectral amplitude coding (SAC-OCDMA) over conventional OCDMA systems is that, when using appropriate detection technique, the multiple access interference (MAI) can totally be canceled. The motivation of this research is to develop new code families to enhance the overall performance of optical OCDMA systems. Four aspects are tackled in this research. Firstly, a comprehensive discussion takes place on all important aspects of existing codes from advantages and disadvantages point of view. Two algorithms are proposed to construct several code families namely Vector Combinatorial (VC). Secondly, a new detection technique based on exclusive-OR (XOR) logic is developed and compared to the reported detection techniques. Thirdly, a software simulation for SAC OCDMA system with the VC families using a commercial optical system, Virtual Photonic Instrument, “VPITM TransmissionMaker 7.1” is conducted. Finally, an extensive investigation to study and characterize the VC-OCDMA in local area network (LAN) is conducted. For the performance analysis, the effects of phase-induced intensity noise (PIIN), shot noise, and thermal noise are considered simultaneously. The performances of the system compared to reported systems were characterized by referring to the signal to noise ratio (SNR), the bit error rate (BER) and the effective power (Psr). Numerical results show that, an acceptable BER of 10−9 was achieved by the VC codes with 120 active users while a much better performance can be achieved when the effective received power Psr > -26 dBm. In particular, the BER can be significantly improved when the VC optimal channel spacing width is carefully selected; best performance occurs at a spacing bandwidth between 0.8 and 1 nm. The simulation results indicate that VC code has a superior performance compared to other reported codes for the same transmission quality. It is also found that for a transmitted power at 0 dBm, the BER specified by eye diagrams patterns are 10-14 and 10-5 for VC and Modified Quadratic Congruence (MQC) codes respectively

Generalizations of All-or-Nothing Transforms and their Application in Secure Distributed Storage

An all-or-nothing transform is an invertible function that maps s inputs to s outputs such that, in the calculation of the inverse, the absence of only one output makes it impossible for an adversary to obtain any information about any single input. In this thesis, we generalize this structure in several ways motivated by different applications, and for each generalization, we provide some constructions. For a particular generalization, where we consider the security of t input blocks in the absence of t output blocks, namely, t-all-or-nothing transforms, we provide two applications. We also define a closeness measure and study structures that are close to t-all-or-nothing transforms. Other generalizations consider the situations where: i) t covers a range of values and the structure maintains its t-all-or-nothingness property for all values of t in that range; ii) the transform provides security for a smaller, yet fixed, number of inputs than the number of absent outputs; iii) the missing output blocks are only from a fixed subset of the output blocks; and iv) the transform generates n outputs so that it can still reconstruct the inputs as long as s outputs are available. In the last case, the absence of n-s+t outputs can protect the security of any t inputs. For each of these transforms, various existence and non-existence results, as well as bounds and equivalence results are presented. We finish with proposing an application of generalization (iv) in secure distributed storage

Applications of perfect difference codes in fiber-optics and wireless optical code-division multiplexing/multiple-access systems

After establishing itself in the radio domain, Spread spectrum code-division multiplexing/multiple-access (CDMA) has seen a recent upsurge in optical domain as well. Due to its fairness, flexibility, service differentiation and increased inherent security, CDMA is proved to be more suitable for the bursty nature of local area networks than synchronous multiplexing techniques like Frequency/Wavelength Division Multiplexing (F/WDM) and Time Division Multiplexing (TDM). In optical domain, CDMA techniques are commonly known as Optical-CDMA (O-CDMA). All optical CDMA systems are plagued with the problem of multiple-access interference (MAI). Spectral amplitude coding (SAC) is one of the techniques used in the literature to deal with the problem of MAI. The choice of spreading code in any CDMA system is another way to ensure the successful recovery of data at the receiving end by minimizing the effect of MAI and it also dictates the hardware design of the encoder and decoder. This thesis focuses on the efficient design of encoding and decoding hardware. Perfect difference codes (PDC) are chosen as spreading sequences due to their good correlation properties. In most of the literature, evaluation of error probability is based on the assumptions of ideal conditions. Such assumptions ignore major physical impairments such as power splitting losses at the multiplexers of transmitters and receivers, and gain losses at the receivers, which may in practice be an overestimate or underestimate of the actual probability of error. This thesis aims to investigate thoroughly with the consideration of practical impairments the applications of PDCs and other spreading sequences in optical communications systems based on spectral-amplitude coding and utilizing codedivision as multiplexing/multiple-access technique. This work begins with a xix general review of optical CDMA systems. An open-ended practical approach has been used to evaluate the actual error probabilities of OCDM/A systems under study. It has been concluded from results that mismatches in the gains of photodetectors, namely avalanche photodiode (APDs), used at the receiver side and uniformity loss in the optical splitters results in the inaccurate calculation of threshold level used to detect the data and can seriously degrade the system bit error rate (BER) performance. This variation in the threshold level can be compensated by employing techniques which maintain a constant interference level so that the decoding architecture does not have to estimate MAI every time to make a data bit decision or by the use of balanced sequences. In this thesis, as a solution to the above problem, a novel encoding and decoding architecture is presented for perfect difference codes based on common zero code technique which maintains a constant interference level at all instants in CDM system and thus relieves the need of estimating interference. The proposed architecture only uses single multiplexer at the transmitters for all users in the system and a simple correlation based receiver for each user. The proposed configuration not only preserves the ability of MAI in Spectral-Amplitude Coding SAC-OCDM system, but also results in a low cost system with reduced complexity. The results show that by using PDCs in such system, the influence of MAI caused by other users can be reduced, and the number of active users can be increased significantly. Also a family of novel spreading sequences are constructed called Manchestercoded Modified Legendre codes (MCMLCs) suitable for SAC based OCDM systems. MCMLCs are designed to be used for both single-rate and Multirate systems. First the construction of MCMLCs is presented and then the bit error rate performance is analyzed. Finally the proposed encoding/decoding architecture utilizing perfect difference codes is applied in wireless infrared environment and the performance is found to be superior to other codes