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In this context our plan is to make our message longer than M bits, so that we have some redundancy in it. We want to use that redundancy to recognize the errors in our message, after it is sent and received, or after it is retrieved from storage, and recover our original message. The remainder of this chapter will consist of proof of this theorem. We will then turn to the second question. The rest of this section consists of philosophical remarks.

A problem of index coding with side information was first considered by Y. Birk and T. Kol (IEEE INFOCOM, 1998). In the present work, a generalization of index coding scheme, where transmitted symbols are subject to errors, is studied. Error-correcting methods for such a scheme, and their parameters, are investigated. In particular, the following question is discussed: given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols $\delta$, what is the shortest length of a linear index code, such that every receiver is able to recover the required information? This question turns out to be a generalization of the problem of finding a shortest-length error-correcting code with a prescribed error-correcting capability in the classical coding theory. The Singleton bound and two other bounds, referred to as the $\alpha$-bound and the $\kappa$-bound, for the optimal length of a linear error-correcting index code (ECIC) are established. For large alphabets, a construction based on concatenation of an optimal index code with an MDS classical code, is shown to attain the Singleton bound. For smaller alphabets, however, this construction may not be optimal. A random construction is also analyzed. It yields another inexplicit bound on the length of an optimal linear ECIC. Finally, the decoding of linear ECIC's is discussed. The syndrome decoding is shown to output the exact message if the weight of the error vector is less or equal to the error-correcting capability of the corresponding ECIC.

Decreasing transistor sizes and lower voltage swings cause two distinct problems for communication in integrated circuits. First, decreasing inter-wire spacing increases interline capacitive coupling, which adversely affects transmission energy and delay. Second, lower voltage swings render the transmission susceptible to various noise sources. Coding can be used to address both these problems. So-called crosstalk-avoidance codes mitigate capacitive coupling, and traditional error-correction codes introduce resilience against channel errors. Unfortunately, crosstalk-avoidance and error-correction codes cannot be combined in a straightforward manner. On the one hand, crosstalk-avoidance encoding followed by error-correction encoding destroys the crosstalk-avoidance property. On the other hand, error-correction encoding followed by crosstalk-avoidance encoding causes the crosstalk-avoidance decoder to fail in the presence of errors. Existing approaches circumvent this difficulty by using additional bus wires to protect the parities generated from the output of the error-correction encoder, and are therefore inefficient. In this work we propose a novel joint crosstalk-avoidance and error-correction coding and decoding scheme that provides higher bus transmission rates compared to existing approaches. Our joint approach carefully embeds the parities such that the crosstalk-avoidance property is preserved. We analyze the rate and minimum distance of the proposed scheme. We also provide a density evolution analysis and predict iterative decoding thresholds for reliable communication under random bus erasures. This density evolution analysis is nonstandard, since the crosstalk-avoidance constraints are inherently nonlinear.

328 pages : 28 cm

An efficient coding circuit is given for the perfect quantum error correction of a single qubit against arbitrary 1-qubit errors within a 5 qubit code. The circuit presented employs a double `classical' code, i.e., one for bit flips and one for phase shifts. An implementation of this coding circuit on an ion-trap quantum computer is described that requires 26 laser pulses. A further circuit is presented requiring only 24 laser pulses, making it an efficient protection scheme against arbitrary 1-qubit errors. In addition, the performance of two error correction schemes, one based on the quantum Zeno effect and the other using standard methods, is compared. The quantum Zeno error correction scheme is found to fail completely for a model of noise based on phase-diffusion.

Many digital communication channels operating at low data rates suffer phase jitter and time varying frequency-offset effects which degrade the signal to noise ratio. Low rate error correction coding is shown to obviate this problem. Experimental results are presented for different low rate coding arrangements and the usefulness of the technique is discussed in relation to both coherent and non-coherent modulation methods. (Author)

This tutorial attempts to provide a frank, step-by-step approach to Reed-Solomon (RS) error correction coding. RS encoding and RS decoding both with and without erasing code symbols are emphasized. There is no need to present rigorous proofs and extreme mathematical detail. Rather, the simple concepts of groups and fields, specifically Galois fields, are presented with a minimum of complexity. Before RS codes are presented, other block codes are presented as a technical introduction into coding. A primitive (15, 9) RS coding example is then completely developed from start to finish, demonstrating the encoding and decoding calculations and a derivation of the famous error-locator polynomial. The objective is to present practical information about Reed-Solomon coding in a manner such that it can be easily understood.

Superdense coding promises increased classical capacity and communication security but this advantage may be undermined by noise in the quantum channel. We present a numerical study of how forward error correction (FEC) applied to the encoded classical message can be used to mitigate against quantum channel noise. By studying the bit error rate under different FEC codes, we identify the unique role that burst errors play in superdense coding, and we show how these can be mitigated against by interleaving the FEC codewords prior to transmission. We conclude that classical FEC with interleaving is a useful method to improve the performance in near-term demonstrations of superdense coding.

Error correction coding can be employed in the design of C3 systems to increase transmission reliability and enhance system effectiveness. The maturation of LSI technology and particularly the ubiquitous microprocessor now allow low-cost simple implementations of error coding hardware. Theoretical fundamentals of linear block codes are reviewed with respect to their suitability to decoder design utilizing 8-bit NMOS microprocessors. Key architectural features of NMOS microprocessors are analyzed to determine those characteristics that are best suited to performing decoding functions. A methodology for evaluating MIL-qualified candidate microprocessors is presented that leads to the selection of a microprocessor for the design of the decoder hardware. (Author)

A problem of index coding with side information was first considered by Y. Birk and T. Kol (IEEE INFOCOM, 1998). In the present work, a generalization of index coding scheme, where transmitted symbols are subject to errors, is studied. Error-correcting methods for such a scheme, and their parameters, are investigated. In particular, the following question is discussed: given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols $\delta$, what is the shortest length of a linear index code, such that every receiver is able to recover the required information? This question turns out to be a generalization of the problem of finding a shortest-length error-correcting code with a prescribed error-correcting capability in the classical coding theory. The Singleton bound and two other bounds, referred to as the $\alpha$-bound and the $\kappa$-bound, for the optimal length of a linear error-correcting index code (ECIC) are established. For large alphabets, a construction based on concatenation of an optimal index code with an MDS classical code, is shown to attain the Singleton bound. For smaller alphabets, however, this construction may not be optimal. A random construction is also analyzed. It yields another implicit bound on the length of an optimal linear ECIC. Further, the problem of error-correcting decoding by a linear ECIC is studied. It is shown that in order to decode correctly the desired symbol, the decoder is required to find one of the vectors, belonging to an affine space containing the actual error vector. The syndrome decoding is shown to produce the correct output if the weight of the error pattern is less or equal to the error-correcting capability of the corresponding ECIC. Finally, the notion of static ECIC, which is suitable for use with a family of instances of an index coding problem, is introduced.

Recent progress in quantum cryptography and quantum computers has given hope to their imminent practical realization. An essential element at the heart of the application of these quantum systems is a quantum error correction scheme. We propose a new technique based on the use of coding in order to detect and correct errors due to imperfect transmission lines in quantum cryptography or memories in quantum computers. We give a particular example of how to detect a decohered qubit in order to transmit or preserve with high fidelity the original qubit.

This thesis focuses on the modeling and simulation of the performance of M-ary modulation techniques. In addition to the most popular used coherent modulation schemes, forward error correction (FEC) coded signals, and fading channels are also considered. The channel is modeled for the two cases of fading and no fading, both with additive white Gaussian noise (AWGN). The effect of barrage noise interference is also considered. Binary phase-shift keying (BPSK), 16-PSK, and 16-QAM (quadrature amplitude modulation) communication systems are simulated and analyzed. For BPSK, both hard decision decoding (HDD) and soft decision decoding (SDD) are considered. For 16-PSK and 16-QAM, only HDD is examined. Simulation results and analytical results are compared. The results show that the modeling and simulation in SystemView, are in excellent agreement with the analytical results.

We design low-complexity error correction coding schemes for channels that introduce different types of errors and erasures: on the one hand, the proposed schemes can successfully deal with symbol errors and erasures, and, on the other hand, they can also successfully handle phased burst errors and erasures.

We design low-complexity error correction coding schemes for channels that introduce different types of errors and erasures: on the one hand, the proposed schemes can successfully deal with symbol errors and erasures, and, on the other hand, they can also successfully handle phased burst errors and erasures.

This report describes the progress made towards the completion of a specific task on error-correcting coding. The proposed research consisted of investigating the use of modulation block codes as the inner code of a concatenated coding system in order to improve the overall space link communications performance. The study proposed to identify and analyze candidate codes that will complement the performance of the overall coding system which uses the interleaved RS (255,223) code as the outer code.

Differential linear network coding (DLNC) is a precoding scheme for information transmission over random linear networks. By using differential encoding and decoding, the conventional approach of lifting, required for inherent channel sounding, can be omitted and in turn higher transmission rates are supported. However, the scheme is sensitive to variations in the network topology. In this paper, we derive an extended DLNC channel model which includes slow network changes. Based on this, we propose and analyze a suitable channel coding scheme matched to the situation at hand using rank-metric convolutional codes.

This thesis focuses on the modeling and simulation of the performance of M-ary modulation techniques. In addition to the most popular used coherent modulation schemes, forward error correction (FEC) coded signals, and fading channels are also considered. The channel is modeled for the two cases of fading and no fading, both with additive white Gaussian noise (AWGN). The effect of barrage noise interference is also considered. Binary phase-shift keying (BPSK), 16-PSK, and 16-QAM (quadrature amplitude modulation) communication systems are simulated and analyzed. For BPSK, both hard decision decoding (HDD) and soft decision decoding (SDD) are considered. For 16-PSK and 16-QAM, only HDD is examined. Simulation results and analytical results are compared. The results show that the modeling and simulation in SystemView, are in excellent agreement with the analytical results.

The problem of error correction in both coherent and noncoherent network coding is considered under an adversarial model. For coherent network coding, where knowledge of the network topology and network code is assumed at the source and destination nodes, the error correction capability of an (outer) code is succinctly described by the rank metric; as a consequence, it is shown that universal network error correcting codes achieving the Singleton bound can be easily constructed and efficiently decoded. For noncoherent network coding, where knowledge of the network topology and network code is not assumed, the error correction capability of a (subspace) code is given exactly by a new metric, called the injection metric, which is closely related to, but different than, the subspace metric of K\"otter and Kschischang. In particular, in the case of a non-constant-dimension code, the decoder associated with the injection metric is shown to correct more errors then a minimum-subspace-distance decoder. All of these results are based on a general approach to adversarial error correction, which could be useful for other adversarial channels beyond network coding.

Design, operation and testing of a table-look-up microprocessor-based (Motorola MC6800) (7,3) Reed-Solomon decoder is presented. Decoder operation is fully illustrated by the use of error and erasure pattern examples. Exhaustive (complete) and random (Monte Carlo) testing is employed to exercise the decoder. Test results are analyzed and conclusions drawn relative to decoder performance. (Author)

In this paper, we present multiple bit error correction coding scheme based on extended Hamming product code combined with type II HARQ using shared resources for on chip interconnect. The shared resources reduce the hardware complexity of the encoder and decoder compared to the existing three stages iterative decoding method for on chip interconnects. The proposed method of decoding achieves 20% and 28% reduction in area and power consumption respectively, with only small increase in decoder delay compared to the existing three stage iterative decoding scheme for multiple bit error correction. The proposed code also achieves excellent improvement in residual flit error rate and up to 58% of total power consumption compared to the other error control schemes. The low complexity and excellent residual flit error rate make the proposed code suitable for on chip interconnection links. 

Index coding is a source coding problem in which a broadcaster seeks to meet the different demands of several users, each of whom is assumed to have some prior information on the data held by the sender. If the sender knows its clients' requests and their side-information sets, then the number of packet transmissions required to satisfy all users' demands can be greatly reduced if the data is encoded before sending. The collection of side-information indices as well as the indices of the requested data is described as an instance of the index coding with side-information (ICSI) problem. The encoding function is called the index code of the instance, and the number of transmissions employed by the code is referred to as its length. The main ICSI problem is to determine the optimal length of an index code for and instance. As this number is hard to compute, bounds approximating it are sought, as are algorithms to compute efficient index codes. Two interesting generalizations of the problem that have appeared in the literature are the subject of this work. The first of these is the case of index coding with coded side information, in which linear combinations of the source data are both requested by and held as users' side-information. The second is the introduction of error-correction in the problem, in which the broadcast channel is subject to noise. In this paper we characterize the optimal length of a scalar or vector linear index code with coded side information (ICCSI) over a finite field in terms of a generalized min-rank and give bounds on this number based on constructions of random codes for an arbitrary instance. We furthermore consider the length of an optimal error correcting code for an instance of the ICCSI problem and obtain bounds on this number, both for the Hamming metric and for rank-metric errors. We describe decoding algorithms for both categories of errors.

Index coding is a source coding problem in which a broadcaster seeks to meet the different demands of several users, each of whom is assumed to have some prior information on the data held by the sender. If the sender knows its clients' requests and their side-information sets, then the number of packet transmissions required to satisfy all users' demands can be greatly reduced if the data is encoded before sending. The collection of side-information indices as well as the indices of the requested data is described as an instance of the index coding with side-information (ICSI) problem. The encoding function is called the index code of the instance, and the number of transmissions employed by the code is referred to as its length. The main ICSI problem is to determine the optimal length of an index code for and instance. As this number is hard to compute, bounds approximating it are sought, as are algorithms to compute efficient index codes. Two interesting generalizations of the problem that have appeared in the literature are the subject of this work. The first of these is the case of index coding with coded side information, in which linear combinations of the source data are both requested by and held as users' side-information. The second is the introduction of error-correction in the problem, in which the broadcast channel is subject to noise. In this paper we characterize the optimal length of a scalar or vector linear index code with coded side information (ICCSI) over a finite field in terms of a generalized min-rank and give bounds on this number based on constructions of random codes for an arbitrary instance. We furthermore consider the length of an optimal error correcting code for an instance of the ICCSI problem and obtain bounds on this number, both for the Hamming metric and for rank-metric errors. We describe decoding algorithms for both categories of errors.

Coherent network error correction is the error-control problem in network coding with the knowledge of the network codes at the source and sink nodes. With respect to a given set of local encoding kernels defining a linear network code, we obtain refined versions of the Hamming bound, the Singleton bound and the Gilbert-Varshamov bound for coherent network error correction. Similar to its classical counterpart, this refined Singleton bound is tight for linear network codes. The tightness of this refined bound is shown by two construction algorithms of linear network codes achieving this bound. These two algorithms illustrate different design methods: one makes use of existing network coding algorithms for error-free transmission and the other makes use of classical error-correcting codes. The implication of the tightness of the refined Singleton bound is that the sink nodes with higher maximum flow values can have higher error correction capabilities.

Probability of bit error (Pb) performance for a turbulent, atmospheric, optical, communications link with convolutional error detection and correction coding was investigated. The codes investigated were all rate 1/2 codes with constraint lengths ranging from 3 to 7. It was assumed that the coded data stream was sufficiently interleaved so that the turbulent channel could be considered memoryless. A theoretical bound on P sub b, given by Viterbi for a memoryless channel, was used to plot curves of P sub b versus the signal intensity of last transmitter. These curves were plotted for three different levels of atmospheric turbulence and three different levels of background light.

This research investigates, via simulation, the bit error probability (BEP) associated with a variable redundancy coding scheme operating in a wireless communications environment. Within a slowly varying (flat fading) Rayleigh channel, adaptive algorithms provide increased throughput over fixed coding implementations. From a family of BCH codes of the same block length (n=63), a code with appropriate redundancy is chosen depending on the receiver's estimation of the current conditions experienced in this channel. Two different decision techniques are compared. The first method statistically evaluates the receiver's input and calculates the signal to noise ratio (Eb/No), while the second method observes the number of corrected errors in recently decoded blocks. With this information, the adaptive system decides to modify the correction ability of the code, and then transmits this decision to the encoder over a low bandwidth feedback channel. The correction ability can be changed on a block by block basis. This algorithm is implemented in software and, therefore, can be optimized for many real world communications systems. The low cost of high speed microprocessors and DSPs allows for the development of a robust adaptive coding system in hardware. The results are compared against fixed coding implementations and show that the adaptive process maintains a better efficiency (eta=k/n) of information rate while keeping the bit error probability near the level obtained by maximum encoding.

Coding; Communications; Engineering; Networks; Information Theory; Algorithms

It was sought to determine if the quality of high speed digital data transmitted over long-haul HF radio circuits could be improved by the use of commercially available forward error correction (FEC) devices. The objective was to improve the bit error rate such that HF radio channels could be reliably used as overflow and restoral circuits between Switching Centers within the AUTODIN System. A study, analysis and test program was conducted.

Codes and coding circuits for automatic error correction in digital computers

With the development of new advanced instruments for remote sensing applications, sensor data will be generated at a rate that not only requires increased onboard processing and storage capability, but imposes demands on the space to ground communication link and ground data management-communication system. Data compression and error control codes provide viable means to alleviate these demands. Two types of data compression have been studied by many researchers in the area of information theory: a lossless technique that guarantees full reconstruction of the data, and a lossy technique which generally gives higher data compaction ratio but incurs some distortion in the reconstructed data. To satisfy the many science disciplines which NASA supports, lossless data compression becomes a primary focus for the technology development. While transmitting the data obtained by any lossless data compression, it is very important to use some error-control code. For a long time, convolutional codes have been widely used in satellite telecommunications. To more efficiently transform the data obtained by the Rice algorithm, it is required to meet the a posteriori probability (APP) for each decoded bit. A relevant algorithm for this purpose has been proposed which minimizes the bit error probability in the decoding linear block and convolutional codes and meets the APP for each decoded bit. However, recent results on iterative decoding of 'Turbo codes', turn conventional wisdom on its head and suggest fundamentally new techniques. During the past several months of this research, the following approaches have been developed: (1) a new lossless data compression algorithm, which is much better than the extended Rice algorithm for various types of sensor data, (2) a new approach to determine the generalized Hamming weights of the algebraic-geometric codes defined by a large class of curves in high-dimensional spaces, (3) some efficient improved geometric Goppa codes for disk memory systems and high-speed mass memory systems, and (4) a tree based approach for data compression using dynamic programming.