Search results for: LDPC codes

Authors: Ni Wenli, He Jing

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Repeat–Accumulate (RA) codes are subclass of LDPC codes with fast encoder structures. In this paper, we consider a nonbinary extension of binary LDPC codes over GF(q) and construct a non-binary RA code and a non-binary QC-LDPC code over GF(2^4), we construct non-binary RA codes with linear encoding method and non-binary QC-LDPC codes with algebraic constructions method. And the BER performance of RA and QC-LDPC codes over GF(q) are compared with BP decoding and by simulation over the Additive White Gaussian Noise (AWGN) channels.

Keywords: non-binary RA codes, QC-LDPC codes, performance comparison, BP algorithm

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Authors: Kassoul Ilyes

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The field of error correcting codes has been revolutionized by the introduction of iteratively decoded codes. Among these, LDPC codes are now a preferred solution thanks to their remarkable performance and low complexity. The binary version of LDPC codes showed even better performance, although it’s decoding introduced greater complexity. This thesis studies the performance of binary LDPC codes using simplified weighted decisions. Information is transported between a transmitter and a receiver by digital transmission systems, either by propagating over a radio channel or also by using a transmission medium such as the transmission line. The purpose of the transmission system is then to carry the information from the transmitter to the receiver as reliably as possible. These codes have not generated enough interest within the coding theory community. This forgetfulness will last until the introduction of Turbo-codes and the iterative principle. Then it was proposed to adopt Pearl's Belief Propagation (BP) algorithm for decoding these codes. Subsequently, Luby introduced irregular LDPC codes characterized by a parity check matrix. And finally, we study simplifications on binary LDPC codes. Thus, we propose a method to make the exact calculation of the APP simpler. This method leads to simplifying the implementation of the system.

Keywords: LDPC, parity check matrix, 5G, BER, SNR

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Authors: Marwa Ben Abdessalem, Amin Zribi, Ammar Bouallègue

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In this paper, a Joint Source Channel coding scheme based on LDPC codes is investigated. We consider two concatenated LDPC codes, one allows to compress a correlated source and the second to protect it against channel degradations. The original information can be reconstructed at the receiver by a joint decoder, where the source decoder and the channel decoder run in parallel by transferring extrinsic information. We investigate the performance of the JSC LDPC code in terms of Bit-Error Rate (BER) in the case of transmission over an Additive White Gaussian Noise (AWGN) channel, and for different source and channel rate parameters. We emphasize how JSC LDPC presents a performance tradeoff depending on the channel state and on the source correlation. We show that, the JSC LDPC is an efficient solution for a relatively low Signal-to-Noise Ratio (SNR) channel, especially with highly correlated sources. Finally, a source-channel rate optimization has to be applied to guarantee the best JSC LDPC system performance for a given channel.

Keywords: AWGN channel, belief propagation, joint source channel coding, LDPC codes

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Authors: Hassan El Ghazi, Mohammed El Jourmi, Tayeb Sadiki, Esmail Ahouzi

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The purpose of this paper is to model and analyze a geostationary satellite communication system based on VSAT network and Multicarrier CDMA system scheme which presents a combination of multicarrier modulation scheme and CDMA concepts. In this study the channel coding strategies (Turbo codes and LDPC codes) are adopted to achieve good performance due to iterative decoding. The envisaged system is examined for a transmission over Multipath channel with use of Ku band in the uplink case. The simulation results are obtained for each different case. The performance of the system is given in terms of Bit Error Rate (BER) and energy per bit to noise power spectral density ratio (Eb/N0). The performance results of designed system shown that the communication system coded with LDPC codes can achieve better error rate performance compared to VSAT MC-CDMA system coded with Turbo codes.

Keywords: satellite communication, VSAT Network, MC-CDMA, LDPC codes, turbo codes, uplink

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Authors: Cinna Soltanpur, Mohammad Ghamari, Behzad Momahed Heravi, Fatemeh Zare

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Low-density parity-check (LDPC) codes have been shown to deliver capacity approaching performance; however, problematic graphical structures (e.g. trapping sets) in the Tanner graph of some LDPC codes can cause high error floors in bit-error-ratio (BER) performance under conventional sum-product algorithm (SPA). This paper presents a serial concatenation scheme to avoid the trapping sets and to lower the error floors of LDPC code. The outer code in the proposed concatenation is the LDPC, and the inner code is a high rate array code. This approach applies an interactive hybrid process between the BCJR decoding for the array code and the SPA for the LDPC code together with bit-pinning and bit-flipping techniques. Margulis code of size (2640, 1320) has been used for the simulation and it has been shown that the proposed concatenation and decoding scheme can considerably improve the error floor performance with minimal rate loss.

Keywords: concatenated coding, low–density parity–check codes, array code, error floors

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Authors: Anurag Saxena, Alkesh Agrawal, Dinesh Kumar

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The next generation mobile communication systems i.e. fourth generation (4G) was developed to accommodate the quality of service and required data rate. This project focuses on multiple access technique proposed in 4G communication systems. It is attempted to demonstrate the IDMA (Interleave Division Multiple Access) technology. The basic principle of IDMA is that interleaver is different for each user whereas CDMA employs different signatures. IDMA inherits many advantages of CDMA such as robust against fading, easy cell planning; dynamic channel sharing and IDMA increase the spectral efficiency and reduce the receiver complexity. In this, performance of IDMA is analyzed using QC-LDPC coding scheme further it is compared with LDPC coding and at last BER is calculated and plotted in MATLAB.

Keywords: 4G, QC-LDPC, CDMA, IDMA

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Authors: Jaskarn Singh Bhullar, Divya Taneja, Manish Gupta, Rajesh Kumar Narula

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The existence conditions of dual containing constacyclic codes have opened a new path for finding quantum maximum distance separable (MDS) codes. Using these conditions parameters of length n=(q²+1)/2 quantum MDS codes were improved. A class of quantum MDS codes of length n=(q²+q+1)/h, where h>1 is an odd prime, have also been constructed having large minimum distance and these codes are new in the sense as these are not available in the literature.

Keywords: hermitian construction, constacyclic codes, cyclotomic cosets, quantum MDS codes, singleton bound

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Authors: Mouna Hemdi, Jamel bel Hadj Tahar

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The increasing development of multimedia applications requiring the simultaneous transport of several different services contributes to the evolution of the need for very high-speed network. In this paper, we propose an effective solution to achieve the very high speed while retaining elements of the optical transmission channel. So our study focuses on error correcting codes that aim for quality improvement on duty. We present a comparison of the quality of service for single channels and integrating the code BCH, RS and LDPC in order to find the best code in the different conditions of the transmission.

Keywords: code error correction, high speed broadband, optical transmission, information systems security

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Authors: Ghania Zerari, Abderrezak Guessoum, Rachid Beguenane

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This paper introduces high-performance architecture for fully parallel stochastic Low-Density Parity-Check (LDPC) field programmable gate array (FPGA) based LDPC decoder. The new approach is designed to decrease the decoding latency and to reduce the FPGA logic utilisation. To accomplish the target logic utilisation reduction, the routing of the proposed sub-variable node (VN) internal memory is designed to utilize one slice distributed RAM. Furthermore, a VN initialization, using the channel input probability, is achieved to enhance the decoder convergence, without extra resources and without integrating the output saturated-counters. The Xilinx FPGA implementation, of IEEE 802.3an standard LDPC code, shows that the proposed decoding approach attain high performance along with reduction of FPGA logic utilisation.

Keywords: low-density parity-check (LDPC) decoder, stochastic decoding, field programmable gate array (FPGA), IEEE 802.3an standard

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Authors: Eusebio Jr. Lina, Ederlina Nocon

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Cyclic codes are a fundamental subclass of linear codes that enjoy a very interesting algebraic structure. The class of skew cyclic codes (or θ-cyclic codes) is a generalization of the notion of cyclic codes. This a very large class of linear codes which can be used to systematically search for codes with good properties. A linear code with complementary dual (LCD code) is a linear code C satisfying C ∩ C^⊥ = {0}. This subclass of linear codes provides an optimum linear coding solution for a two-user binary adder channel and plays an important role in countermeasures to passive and active side-channel analyses on embedded cryptosystems. This paper aims to identify LCD codes from the class of skew cyclic codes. Let F_q be a finite field of order q, and θ be an automorphism of F_q. Some conditions for a skew cyclic code to be LCD were given. To this end, the properties of a noncommutative skew polynomial ring F_q[x, θ] of automorphism type were revisited, and the algebraic structure of skew cyclic code using its skew polynomial representation was examined. Using the result that skew cyclic codes are left ideals of the ring F_q[x, θ]/〈x^n-1〉, a characterization of a skew cyclic LCD code of length n was derived. A necessary condition for a skew cyclic code to be LCD was also given.

Keywords: LCD cyclic codes, skew cyclic LCD codes, skew cyclic complementary dual codes, theta-cyclic codes with complementary duals

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Authors: Jing Li, Xiuli Li

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This paper studies a special class of linear codes, called skew cyclic codes, over the ring R= F_q+uF_q+…+u^k-1F_q, where q is a prime power. A Gray map ɸ from R to F_q and a Gray map ɸ' from Rⁿ to Fⁿ_q are defined, as well as an automorphism Θ over R. It is proved that the images of skew cyclic codes over R under map ɸ' and Θ are cyclic codes over F_q, and they still keep the dual relation.

Keywords: skew cyclic code, gray map, automorphism, cyclic code

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Authors: Ibrahim Özbek, Erdoğan Mehmet Özkan

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In this study, we generalize (k,n) threshold secret sharing scheme on the study Ozbek and Siap to the codes over the ring Fq+ αFq. In this way, it is mentioned that the method obtained in that article can also be used on codes over rings, and new advantages to be obtained. The method of securely sharing the key in cryptography, which Shamir first systematized and Massey carried over to codes, became usable for all error-correcting codes. The firewall of this scheme is based on the hardness of the syndrome decoding problem. Also, an open study area is left for those working for other rings and code classes. All codes that correct errors with this method have been the working area of this method.

Keywords: secret sharing scheme, linear codes, algebra, finite rings

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Authors: Arun Moorthy

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Quantum computing promises to provide algorithmic speedups for a number of tasks; however, similar to classical computing, effective error-correcting codes are needed. Current quantum computers require costly equipment to control each particle, so having fewer particles to control is ideal. Although traditional quantum computers are built using qubits (2-level systems), qudits (more than 2-levels) are appealing since they can have an equivalent computational space using fewer particles, meaning fewer particles need to be controlled. Currently, qudit quantum error-correction codes are available for different level qudit systems; however, these codes have sometimes overly specific constraints. When building a qudit system, it is important for researchers to have access to many codes to satisfy their requirements. This project addresses two methods to increase the number of quantum error correcting codes available to researchers. The first method is generating new codes for a given set of parameters. The second method is generating new error-correction codes by using existing codes as a starting point to generate codes for another level (i.e., a 5-level system code on a 2-level system). So, this project builds a website that researchers can use to generate new error-correction codes or codes based on existing codes.

Keywords: qudit, error correction, quantum, qubit

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Authors: Nikolai Krainiukov, Boris Melnikov

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We study the properties of maximal prefix codes. The codes have many applications in computer science, theory of formal languages, data processing and data classification. Practical application is based on the representation of the maximal prefix codes as a sequence of words in a specific order. Our approach to study uses finite state automata (so-called flower automata) for the representation of prefix codes. An important task is the decomposition of prefix codes into prime prefix codes (factors). We discuss the properties of such prefix code decompositions. A linear time algorithm is designed which find the prime decomposition. To verify the correctness of the proposed algorithms, we implemented a system computer algebra GAP.

Keywords: maximal prefix code, regular languages, flower automata, prefix code decomposing

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Authors: Skezeer John B. Paz, Ederlina G. Nocon

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Finding an optimal binary linear code is a central problem in coding theory. A binary linear code C = [n, k, d] is called optimal if there is no linear code with higher minimum distance d given the length n and the dimension k. There are bounds giving limits for the minimum distance d of a linear code of fixed length n and dimension k. The lower bound which can be taken by construction process tells that there is a known linear code having this minimum distance. The upper bound is given by theoretic results such as Griesmer bound. One way to find an optimal binary linear code is to make the lower bound of d equal to its higher bound. That is, to construct a binary linear code which achieves the highest possible value of its minimum distance d, given n and k. Some optimal binary linear codes were presented by Andries Brouwer in his published table on bounds of the minimum distance d of binary linear codes for 1 ≤ n ≤ 256 and k ≤ n. This was further improved by Markus Grassl by giving a detailed construction process for each code exhibiting the lower bound. In this paper, we construct new optimal binary linear codes by using some construction processes on existing binary linear codes. Particularly, we developed an algorithm applied to the codes already constructed to extend the list of optimal binary linear codes up to 257 ≤ n ≤ 300 for k ≤ 7.

Keywords: bounds of linear codes, Griesmer bound, construction of linear codes, optimal binary linear codes

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Authors: Jamal Laaouine, Mohammed Elhassani Charkani

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Let p be a prime and let b be an integer. MDS b-symbol codes are a direct generalization of MDS codes. The γ-constacyclic codes of length pˢ over the finite commutative chain ring Fₚm [u]/ < u² > had been classified into four distinct types, where is a nonzero element of the field Fₚm. Let C₃ be a code of Type 3. In this paper, we obtain the b-symbol distance db(C₃) of the code C₃. Using this result, necessary and sufficient conditions under which C₃ is an MDS b-symbol code are given.

Keywords: constacyclic code, repeated-root code, maximum distance separable, MDS codes, b-symbol distance, finite chain rings

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Authors: Gurjit Kaur, Neena Gupta

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In this paper, we have analyzed and compared the performance of various coding schemes. The basic ID prime sequence codes are unique in only dimension, i.e. time slots, whereas 2D coding techniques are not unique by their time slots but with their wavelengths also. In this research, we have evaluated and compared the performance of 1D and 2D coding techniques constructed using prime sequence coding pattern for Optical Code Division Multiple Access (OCDMA) system on a single platform. Analysis shows that 2D prime code supports lesser number of active users than 1D codes, but they are having large code family and are the most secure codes compared to other codes. The performance of all these codes is analyzed on basis of number of active users supported at a Bit Error Rate (BER) of 10-9.

Keywords: CDMA, OCDMA, BER, OOC, PC, EPC, MPC, 2-D PC/PC, λc, λa

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Authors: Gurjit Kaur, Shashank Johri, Arpit Mehrotra

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In this paper we have analyzed and compared the performance of various coding schemes. The basic ID prime sequence codes are unique in only dimension i.e. time slots whereas 2D coding techniques are not unique by their time slots but with their wavelengths also. In this research we have evaluated and compared the performance of 1D and 2D coding techniques constructed using prime sequence coding pattern for OCDMA system on a single platform. Results shows that 1D Extended Prime Code (EPC) can support more number of active users compared to other codes but at the expense of larger code length which further increases the complexity of the code. Modified Prime Code (MPC) supports lesser number of active users at λc=2 but it has a lesser code length as compared to 1D prime code. Analysis shows that 2D prime code supports lesser number of active users than 1D codes but they are having large code family and are the most secure codes compared to other codes. The performance of all these codes is analyzed on basis of number of active users supported at a Bit Error Rate (BER) of 10-9.

Keywords: CDMA, OCDMA, BER, OOC, PC, EPC, MPC, 2-D PC/PC, λc, λa

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Authors: Arif Armagan Gozutok, Alper Kule, Burak Tos, Selman Demirel

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Wireless communication chain requires effective ways to keep throughput efficiency high while it suffers location-dependent, time-varying burst errors. Several techniques are developed in order to assure that the receiver recovers the transmitted information without errors. The most fundamental approaches are error checking and correction besides re-transmission of the non-acknowledged packets. In this paper, stop & wait (SAW) and chase combined (CC) hybrid automated repeat request (HARQ) protocols are compared and analyzed in terms of throughput and average delay for the usage of low earth orbit (LEO) mega-constellations case. Several assumptions and technological implementations are considered as well as usage of low-density parity check (LDPC) codes together with several constellation orbit configurations.

Keywords: HARQ, LEO, satellite constellation, throughput

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Authors: Eleanor L. Kutschera

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This study explores the possible impact that QR codes play in helping individuals make more sustainable choices regarding apparel consumption. Data was collected via an online survey to ascertain individuals’ knowledge, attitudes, and behaviors with regard to QR codes and how this impacts their decisions to purchase apparel. Results from 250 participants provide both qualitative and quantitative data that provide valuable information regarding consumers’ use of QR codes and more sustainable purchases. Specifically, results indicate that QR codes are currently under-utilized in the apparel industry but have the potential to generate more environmentally conscious purchases. Also, results posit that while the cost of the item is the most influential factor in purchasing sustainable garments, other factors such as how, where, and what it is made of are in the middle, along with the company’s story/inspiration for creation have an impact. Moreover, participants posit the use of QR codes could make them more informed and empowered consumers, and they would be more likely to make purchases that are better for the environment. Participants’ qualitative responses provide useful incentives that could increase their future sustainable purchases. Finally, this study touches on the study’s limitations, implications, and future direction of research.

Keywords: digital ID, QR codes, environmental consciousness, sustainability, fashion industry, apparel consumption

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Authors: Jaswinder Singh

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In this paper, comparative analysis is carried out to study the performance of wavelength/time/space optical CDMA codes using two well-known formats; those are CSRZ and DPSK-RZ using RSoft’s OptSIM. The analysis is carried out under the real-like scenario considering the presence of various non-linear effects such as XPM, SPM, SRS, SBS and FWM. Fiber dispersion and the multiple access interference are also considered. The codes used in this analysis are 3-D wavelength/time/space codes. These are converted into 2-D wavelength-time codes so that their requirement of space couplers and fiber ribbons is eliminated. Under the conditions simulated, this is found that CSRZ performs better than DPSK-RZ for fiber-optic CDMA applications.

Keywords: Optical CDMA, Multiple access interference (MAI), CSRZ, DPSK-RZ

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Authors: Alla Levina, Sergey Taranov

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The classical approach to the providing noise immunity and integrity of information that process in computing devices and communication channels is to use linear codes. Linear codes have fast and efficient algorithms of encoding and decoding information, but this codes concentrate their detect and correct abilities in certain error configurations. To protect against any configuration of errors at predetermined probability can robust codes. This is accomplished by the use of perfect nonlinear and almost perfect nonlinear functions to calculate the code redundancy. The paper presents the error-correcting coding scheme using biorthogonal wavelet transform. Wavelet transform applied in various fields of science. Some of the wavelet applications are cleaning of signal from noise, data compression, spectral analysis of the signal components. The article suggests methods for constructing linear codes based on wavelet decomposition. For developed constructions we build generator and check matrix that contain the scaling function coefficients of wavelet. Based on linear wavelet codes we develop robust codes that provide uniform protection against all errors. In article we propose two constructions of robust code. The first class of robust code is based on multiplicative inverse in finite field. In the second robust code construction the redundancy part is a cube of information part. Also, this paper investigates the characteristics of proposed robust and linear codes.

Keywords: robust code, linear code, wavelet decomposition, scaling function, error masking probability

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Authors: Chu Chen, Giorgio Gotti, Tony Kang, Michael Wolfe

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This study examines the role of codes of ethics in reducing the extent to which managers’ act opportunistically in reporting earnings. Corporate codes of ethics, by clarifying the boundaries of ethical corporate behaviors and making relevant social norms more salient, have the potential to deter managers from engaging in opportunistic financial reporting practices. In a sample of international companies, we find that the quality of corporate codes of ethics is associated with higher earnings quality, i.e., lower discretionary accruals. Our results are confirmed for a subsample of firms more likely to be engaging in opportunistic reporting behavior, i.e., firms that just meet or beat analysts’ forecasts. Further, codes of ethics play a greater role in reducing earnings management for firms in countries with weaker investor protection mechanisms. Our results suggest that corporate codes of ethics can be a viable alternative to country-level investor protection mechanisms in curbing aggressive reporting behaviors.

Keywords: corporate ethics policy, code of ethics, business ethics, earnings discretion, accruals

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Authors: Moshe Davidian, Shlomo Mark, Yotam Lurie

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With the rapid development of technology and the concomitant growth in the capability of Artificial Intelligence (AI) systems and their power, the ethical challenges involved in these systems are also evolving and increasing. In recent years, various organizations, including governments, international institutions, professional societies, civic organizations, and commercial companies, have been choosing to address these various challenges by publishing ethical codes for AI systems. However, despite the apparent agreement that AI should be “ethical,” there is debate about the definition of “ethical artificial intelligence.” This study investigates the various AI ethical codes and their key ethical values. From the vast collection of codes that exist, it analyzes and compares 25 ethical codes that were found to be representative of different types of organizations. In addition, as part of its literature review, the study overviews data collected in three recent reviews of AI codes. The results of the analyses demonstrate a convergence around seven key ethical values. However, the key finding is that the different AI ethical codes eventually reflect the type of organization that designed the code; i.e., the organizations’ role as regulator, user, or developer affects the view of what ethical AI is. The results show a relationship between the organization’s role and the dominant values in its code. The main contribution of this study is the development of a list of the key values for all AI systems and specific values that need to impact the development and design of AI systems, but also allowing for differences according to the organization for which the system is being developed. This will allow an analysis of AI values in relation to stakeholders.

Keywords: artificial intelligence, ethical codes, principles, values

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Authors: Marsden Jacques, Dennis Wong

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A weak order is a way to rank n objects where ties are allowed. In this talk, we present a recursive framework to generate Gray codes for weak orders. We then describe a simple algorithm based on the framework that generates 2-Gray codes for weak orders in constant amortized time per string. This framework can easily be modified to generate other Gray codes for weak orders. We provide an example on using the framework to generate the first Shift Gray code for weak orders, also in constant amortized time, where consecutive strings differ by a shift or a symbol change.

Keywords: weak order, Cayley permutation, Gray code, shift Gray code

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Authors: Carlo Gadingan

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Swardspeak, as a language, highlights the exclusive identity of the Filipino gay men and the oppression they are confronted in the society. This paper presents a morphological analysis of swardspeak in the Philippines. Specifically, it aims to find out the common morphological processes involved in the construction of codes that may unmask the nature of swardspeak as a language. 30 purposively selected expert users of swardspeak from Luzon, Visayas, and Mindanao were asked to codify 30 natural words through the Facebook Messenger application. The results of the structural analysis affirm that swardspeak follows no specific rules revealing complicated combinations of clipping/stylized clipping, borrowing, connotation through images, connotation through actions, connotation through sounds, affixation, repetition, substitution, and simple reversal. Moreover, it was also found out that most of these word formation processes occur in all word classes which indicate that swardspeak is very unpredictable. Although different codes are used for the same words, there are still codes that are really common to all homosexuals and these are Chaka (ugly), Crayola (cry), and Aida (referring to a person with AIDS). Hence, the prevailing word formation processes explored may be termed as observed time-specific patterns because the codes documented in this study may turn obsolete and may be replaced with novel ones in a matter of weeks to month, knowing the creativity of homosexuals and the multiplicity of societal resources which can be used to make the codes more opaque and more confusing for non-homosexuals.

Keywords: codes, homosexuals, morphological processes, swardspeak

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Authors: Soner Guler, Eylem Guzel, Mustafa Gülen

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Recently, concrete-filled steel tube columns are highly popular in high-rise buildings. The main aim of this study is to evaluate the axial load capacities of circular high strength concrete-filled steel tube columns according to Eurocode 4 (EC4) and American Concrete Institute (ACI) design codes. The axial load capacities of fifteen concrete-filled steel tubes stub columns were compared with design codes EU4 and ACI. The results showed that the EC4 overestimate the axial load capacity for all the specimens.

Keywords: concrete-filled steel tube column, axial load capacity, Eurocode 4, ACI design codes

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Authors: Amin Saeidi

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Using a representation-theoretic approach and considering G to be a finite primitive permutation group of degree n, our aim is to determine linear codes of length n that admit G as a permutation automorphism group. We can show that in some cases, every binary linear code admitting G as a permutation automorphism group is a submodule of a permutation module defined by a primitive action of G. As an illustration of the method, we consider the sporadic simple group M₁₁ and the unitary group U(3,3). We also construct some point- and block-primitive 1-designs from the supports of some codewords of the codes in the discussion.

Keywords: linear code, permutation representation, support design, simple group

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Authors: Catarina Cruz, Ana Breda

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Tiling problems have been capturing the attention of many mathematicians due to their real-life applications. In this study, we deal with tilings of Zⁿ by Lee spheres, where n is a positive integer number, being these tilings related with error correcting codes on the transmission of information over a noisy channel. We focus our attention on the question ‘for what values of n and r does the n-dimensional Lee sphere of radius r tile Zⁿ?’. It seems that the n-dimensional Lee sphere of radius r does not tile Zⁿ for n ≥ 3 and r ≥ 2. Here, we prove that is not possible to tile Z⁷ with Lee spheres of radius 2 presenting a proof based on a combinatorial method and faithful to the geometric idea of the problem. The non-existence of such tilings has been studied by several authors being considered the most difficult cases those in which the radius of the Lee spheres is equal to 2. The relation between these tilings and error correcting codes is established considering the center of a Lee sphere as a codeword and the other elements of the sphere as words which are decoded by the central codeword. When the Lee spheres of radius r centered at elements of a set M ⊂ Zⁿ tile Zⁿ, M is a perfect r-error correcting Lee code of word length n over Z, denoted by PL(n, r). Our strategy to prove the non-existence of PL(7, 2) codes are based on the assumption of the existence of such code M. Without loss of generality, we suppose that O ∈ M, where O = (0, ..., 0). In this sense and taking into account that we are dealing with Lee spheres of radius 2, O covers all words which are distant two or fewer units from it. By the definition of PL(7, 2) code, each word which is distant three units from O must be covered by a unique codeword of M. These words have to be covered by codewords which dist five units from O. We prove the non-existence of PL(7, 2) codes showing that it is not possible to cover all the referred words without superposition of Lee spheres whose centers are distant five units from O, contradicting the definition of PL(7, 2) code. We achieve this contradiction by combining the cardinality of particular subsets of codewords which are distant five units from O. There exists an extensive literature on codes in the Lee metric. Here, we present a new approach to prove the non-existence of PL(7, 2) codes.

Keywords: Golomb-Welch conjecture, Lee metric, perfect Lee codes, tilings

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Authors: Chieh-Fu Chang, Wan-Hsin Hsieh

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We have proposed a novel scheme— iterative word-extension (IWE) to enhance the cliff effect of Reed-Solomon codes regarding the error performance at a specific Eb/N0. The scheme can be readily extended to block codes and the important properties of IWE are further investigated here. In order to select proper block codes specifying the desired cliff Eb/N0, the associated features of IWE are explored. These properties and features grant IWE ability to enhance security regarding the received Eb/N0 in physical layer so that IWE scheme can cooperate with the traditional presentation layer approach — cryptography, to meet the secure requirements in diverse applications. The features and feasibility of IWE scheme in satellite communication are finally discussed.

Keywords: security, IWE, cliff effect, space communications

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