An upper bound for the number of certain error correcting codes /

"August 9, 1954"

TAP for parity check error correcting codes

We employ the methods presented in the previous chapter for decoding corrupted codewords, encoded using sparse parity check error correcting codes. We show the similarity between the equations derived from the TAP approach and those obtained from belief propagation, and examine their performance as practical decoding methods.

Statistical physics of error-correcting codes

In this thesis we use statistical physics techniques to study the typical performance of four families of error-correcting codes based on very sparse linear transformations: Sourlas codes, Gallager codes, MacKay-Neal codes and Kanter-Saad codes. We map the decoding problem onto an Ising spin system with many-spins interactions. We then employ the replica method to calculate averages over the quenched disorder represented by the code constructions, the arbitrary messages and the random noise vectors. We find, as the noise level increases, a phase transition between successful decoding and failure phases. This phase transition coincides with upper bounds derived in the information theory literature in most of the cases. We connect the practical decoding algorithm known as probability propagation with the task of finding local minima of the related Bethe free-energy. We show that the practical decoding thresholds correspond to noise levels where suboptimal minima of the free-energy emerge. Simulations of practical decoding scenarios using probability propagation agree with theoretical predictions of the replica symmetric theory. The typical performance predicted by the thermodynamic phase transitions is shown to be attainable in computation times that grow exponentially with the system size. We use the insights obtained to design a method to calculate the performance and optimise parameters of the high performance codes proposed by Kanter and Saad.

One-time signature scheme from syndrome decoding over generic error-correcting codes

We describe a one-time signature scheme based on the hardness of the syndrome decoding problem, and prove it secure in the random oracle model. Our proposal can be instantiated on general linear error correcting codes, rather than restricted families like alternant codes for which a decoding trapdoor is known to exist. (C) 2010 Elsevier Inc. All rights reserved,

Erasure error correcting codes applied to DTN communications.

The space environment has always been one of the most challenging for communications, both at physical and network layer. Concerning the latter, the most common challenges are the lack of continuous network connectivity, very long delays and relatively frequent losses. Because of these problems, the normal TCP/IP suite protocols are hardly applicable. Moreover, in space scenarios reliability is fundamental. In fact, it is usually not tolerable to lose important information or to receive it with a very large delay because of a challenging transmission channel. In terrestrial protocols, such as TCP, reliability is obtained by means of an ARQ (Automatic Retransmission reQuest) method, which, however, has not good performance when there are long delays on the transmission channel. At physical layer, Forward Error Correction Codes (FECs), based on the insertion of redundant information, are an alternative way to assure reliability. On binary channels, when single bits are flipped because of channel noise, redundancy bits can be exploited to recover the original information. In the presence of binary erasure channels, where bits are not flipped but lost, redundancy can still be used to recover the original information. FECs codes, designed for this purpose, are usually called Erasure Codes (ECs). It is worth noting that ECs, primarily studied for binary channels, can also be used at upper layers, i.e. applied on packets instead of bits, offering a very interesting alternative to the usual ARQ methods, especially in the presence of long delays. A protocol created to add reliability to DTN networks is the Licklider Transmission Protocol (LTP), created to obtain better performance on long delay links. The aim of this thesis is the application of ECs to LTP.

Cyclic spot-error-correcting codes.

Vita.

Error-correcting codes on a Bethe-like lattice

We analyse Gallager codes by employing a simple mean-field approximation that distorts the model geometry and preserves important interactions between sites. The method naturally recovers the probability propagation decoding algorithm as a minimization of a proper free-energy. We find a thermodynamical phase transition that coincides with information theoretical upper-bounds and explain the practical code performance in terms of the free-energy landscape.

Belief revision and small loops in Gallager-type error-correcting codes

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Iterative Phase Optimization of Elementary Quantum Error Correcting Codes

Performing experiments on small-scale quantum computers is certainly a challenging endeavor. Many parameters need to be optimized to achieve high-fidelity operations. This can be done efficiently for operations acting on single qubits, as errors can be fully characterized. For multiqubit operations, though, this is no longer the case, as in the most general case, analyzing the effect of the operation on the system requires a full state tomography for which resources scale exponentially with the system size. Furthermore, in recent experiments, additional electronic levels beyond the two-level system encoding the qubit have been used to enhance the capabilities of quantum-information processors, which additionally increases the number of parameters that need to be controlled. For the optimization of the experimental system for a given task (e.g., a quantum algorithm), one has to find a satisfactory error model and also efficient observables to estimate the parameters of the model. In this manuscript, we demonstrate a method to optimize the encoding procedure for a small quantum error correction code in the presence of unknown but constant phase shifts. The method, which we implement here on a small-scale linear ion-trap quantum computer, is readily applicable to other AMO platforms for quantum-information processing.

Error correcting codes for visible light communication systems

Over the past few years, the number of wireless networks users has been increasing. Until now, Radio-Frequency (RF) used to be the dominant technology. However, the electromagnetic spectrum in these region is being saturated, demanding for alternative wireless technologies. Recently, with the growing market of LED lighting, the Visible Light Communications has been drawing attentions from the research community. First, it is an eficient device for illumination. Second, because of its easy modulation and high bandwidth. Finally, it can combine illumination and communication in the same device, in other words, it allows to implement highly eficient wireless communication systems. One of the most important aspects in a communication system is its reliability when working in noisy channels. In these scenarios, the received data can be afected by errors. In order to proper system working, it is usually employed a Channel Encoder in the system. Its function is to code the data to be transmitted in order to increase system performance. It commonly uses ECC, which appends redundant information to the original data. At the receiver side, the redundant information is used to recover the erroneous data. This dissertation presents the implementation steps of a Channel Encoder for VLC. It was consider several techniques such as Reed-Solomon and Convolutional codes, Block and Convolutional Interleaving, CRC and Puncturing. A detailed analysis of each technique characteristics was made in order to choose the most appropriate ones. Simulink models were created in order to simulate how diferent codes behave in diferent scenarios. Later, the models were implemented in a FPGA and simulations were performed. Hardware co-simulations were also implemented to faster simulation results. At the end, diferent techniques were combined to create a complete Channel Encoder capable of detect and correct random and burst errors, due to the usage of a RS(255,213) code with a Block Interleaver. Furthermore, after the decoding process, the proposed system can identify uncorrectable errors in the decoded data due to the CRC-32 algorithm.

Performance Analysis of Topological Quantum Error Correcting Codes

In the last few years there has been a great development of techniques like quantum computers and quantum communication systems, due to their huge potentialities and the growing number of applications. However, physical qubits experience a lot of nonidealities, like measurement errors and decoherence, that generate failures in the quantum computation. This work shows how it is possible to exploit concepts from classical information in order to realize quantum error-correcting codes, adding some redundancy qubits. In particular, the threshold theorem states that it is possible to lower the percentage of failures in the decoding at will, if the physical error rate is below a given accuracy threshold. The focus will be on codes belonging to the family of the topological codes, like toric, planar and XZZX surface codes. Firstly, they will be compared from a theoretical point of view, in order to show their advantages and disadvantages. The algorithms behind the minimum perfect matching decoder, the most popular for such codes, will be presented. The last section will be dedicated to the analysis of the performances of these topological codes with different error channel models, showing interesting results. In particular, while the error correction capability of surface codes decreases in presence of biased errors, XZZX codes own some intrinsic symmetries that allow them to improve their performances if one kind of error occurs more frequently than the others.

Time course of error detection and correction in humans: neurophysiological evidence.

Using event-related brain potentials, the time course of error detection and correction was studied in healthy human subjects. A feedforward model of error correction was used to predict the timing properties of the error and corrective movements. Analysis of the multichannel recordings focused on (1) the error-related negativity (ERN) seen immediately after errors in response- and stimulus-locked averages and (2) on the lateralized readiness potential (LRP) reflecting motor preparation. Comparison of the onset and time course of the ERN and LRP components showed that the signs of corrective activity preceded the ERN. Thus, error correction was implemented before or at least in parallel with the appearance of the ERN component. Also, the amplitude of the ERN component was increased for errors, followed by fast corrective movements. The results are compatible with recent views considering the ERN component as the output of an evaluative system engaged in monitoring motor conflict.

Generator Matrix Based Search for Extremal Self-Dual Binary Error-Correcting Codes

Self-dual doubly even linear binary error-correcting codes, often referred to as Type II codes, are codes closely related to many combinatorial structures such as 5-designs. Extremal codes are codes that have the largest possible minimum distance for a given length and dimension. The existence of an extremal (72,36,16) Type II code is still open. Previous results show that the automorphism group of a putative code C with the aforementioned properties has order 5 or dividing 24. In this work, we present a method and the results of an exhaustive search showing that such a code C cannot admit an automorphism group Z6. In addition, we present so far unpublished construction of the extended Golay code by P. Becker. We generalize the notion and provide example of another Type II code that can be obtained in this fashion. Consequently, we relate Becker's construction to the construction of binary Type II codes from codes over GF(2^r) via the Gray map.

Spectrum Distribution in Cognitive Radio: Error Correcting Codes Perspective

Cognitive radio is a growing zone in wireless communication which offers an opening in complete utilization of incompetently used frequency spectrum: deprived of crafting interference for the primary (authorized) user, the secondary user is indorsed to use the frequency band. Though, scheming a model with the least interference produced by the secondary user for primary user is a perplexing job. In this study we proposed a transmission model based on error correcting codes dealing with a countable number of pairs of primary and secondary users. However, we obtain an effective utilization of spectrum by the transmission of the pairs of primary and secondary users' data through the linear codes with different given lengths. Due to the techniques of error correcting codes we developed a number of schemes regarding an appropriate bandwidth distribution in cognitive radio.