Elements Of Algebraic Coding Theory PdfCoding theory - Wikipedia, the free encyclopedia. A two- dimensional visualisation of the Hamming distance, a critical measure in coding theory. Coding theory is the study of the properties of codes and their fitness for a specific application. Codes are used for data compression, cryptography, error- correction, and networking. Codes are studied by various scientific disciplines. This typically involves the removal of redundancy and the correction or detection of errors in the transmitted data. There are four types of coding. For example, Zip data compression makes data files smaller to reduce Internet traffic. Data compression and error correction may be studied in combination. Error correction adds extra data bits to make the transmission of data more robust to disturbances present on the transmission channel. The ordinary user may not be aware of many applications using error correction. A typical music CD uses the Reed- Solomon code to correct for scratches and dust. In this application the transmission channel is the CD itself. Cell phones also use coding techniques to correct for the fading and noise of high frequency radio transmission. Data modems, telephone transmissions, and NASA all employ channel coding techniques to get the bits through, for example the turbo code and LDPC codes. At the Dawn of the Theory of Codes. Gaspard-Clair-Fran<. Letqbea prime powerandsupposeF = IFq is a finite field of qelements. Let n = qm 1, Q = qm; denoteby. Introduction to Algebraic Coding Theory Supplementary material for Math 336 Cornell University Sarah A. Spence Contents 1 Introduction 1 2 Basics 2. The term algebraic coding theory denotes the sub-field of coding theory where the properties of codes are expressed in algebraic terms and then further researched. The theory of coding uses the N-dimensional sphere model. Algebraic Coding Theory Honors Thesis Ahsan Ashraf Expected date of graduation: May 15, 2010 Dr. Arnold Feldman Franklin & Marshall College Department of Mathematics and Computer Science May 13, 2010 Abstract. Algebraic Coding Theory. Description: Access Statement. This item appears in the following Collection(s) F&M Theses Collection Algebraic Coding Theory PDF Free Download, Reviews, Read Online, ISBN: 9814635898, By Elwyn R Berlekamp. Algebraic coding theory. Improved decoding of Reed-Solomon and algebraic-geometric codes. To view the rest of this content please follow the download PDF link above. Introduction to Algebraic Coding Theory With Gap Fall 2006 Sarah Spence Adams. History of coding theory. This work focuses on the problem of how best to encode the information a sender wants to transmit. In this fundamental work he used tools in probability theory, developed by Norbert Wiener, which were in their nascent stages of being applied to communication theory at that time. Shannon developed information entropy as a measure for the uncertainty in a message while essentially inventing the field of information theory. The binary Golay code was developed in 1. More specifically, it is an error- correcting code capable of correcting up to three errors in each 2. Richard Hamming won the Turing Award in 1. Bell Labs in numerical methods, automatic coding systems, and error- detecting and error- correcting codes. He invented the concepts known as Hamming codes, Hamming windows, Hamming numbers, and Hamming distance. Source coding. Basically, source codes try to reduce the redundancy present in the source, and represent the source with fewer bits that carry more information. Data compression which explicitly tries to minimize the average length of messages according to a particular assumed probability model is called entropy encoding. Various techniques used by source coding schemes try to achieve the limit of Entropy of the source. In particular, no source coding scheme can be better than the entropy of the source. Example. Source coding removes all data superfluous to the need of the transmitter, decreasing the bandwidth required for transmission. Channel coding. While not mutually exclusive, performance in these areas is a trade off. So, different codes are optimal for different applications. The needed properties of this code mainly depend on the probability of errors happening during transmission. In a typical CD, the impairment is mainly dust or scratches. Thus codes are used in an interleaved manner. Suppose we take a block of data bits (representing sound) and send it three times. At the receiver we will examine the three repetitions bit by bit and take a majority vote. The twist on this is that we don't merely send the bits in order. The block of data bits is first divided into 4 smaller blocks. Then we cycle through the block and send one bit from the first, then the second, etc. This is done three times to spread the data out over the surface of the disk. In the context of the simple repeat code, this may not appear effective. However, there are more powerful codes known which are very effective at correcting the . Deep space communications are limited by the thermal noise of the receiver which is more of a continuous nature than a bursty nature. Likewise, narrowband modems are limited by the noise, present in the telephone network and also modeled better as a continuous disturbance. The high frequencies used can cause rapid fading of the signal even if the receiver is moved a few inches. Again there are a class of channel codes that are designed to combat fading. There are block codes that are not linear, but it is difficult to prove that a code is a good one without this property. In two dimensions, it is easy to visualize. Take a bunch of pennies flat on the table and push them together. The result is a hexagon pattern like a bee's nest. But block codes rely on more dimensions which cannot easily be visualized. The powerful (2. 4,1. Golay code used in deep space communications uses 2. If used as a binary code (which it usually is) the dimensions refer to the length of the codeword as defined above. The theory of coding uses the N- dimensional sphere model. For example, how many pennies can be packed into a circle on a tabletop, or in 3 dimensions, how many marbles can be packed into a globe. Other considerations enter the choice of a code. For example, hexagon packing into the constraint of a rectangular box will leave empty space at the corners. As the dimensions get larger, the percentage of empty space grows smaller. But at certain dimensions, the packing uses all the space and these codes are the so- called . The only nontrivial and useful perfect codes are the distance- 3 Hamming codes with parameters satisfying (2r . First we pack the pennies in a rectangular grid. Each penny will have 4 near neighbors (and 4 at the corners which are farther away). In a hexagon, each penny will have 6 near neighbors. When we increase the dimensions, the number of near neighbors increases very rapidly. The result is the number of ways for noise to make the receiver choose a neighbor (hence an error) grows as well. This is a fundamental limitation of block codes, and indeed all codes. It may be harder to cause an error to a single neighbor, but the number of neighbors can be large enough so the total error probability actually suffers. For example, the syndrome- coset uniqueness property of linear block codes is used in trellis shaping. This same property is used in sensor networks for distributed source coding. Convolutional codes. This is like convolution used in LTI systems to find the output of a system, when you know the input and impulse response. So we generally find the output of the system convolutional encoder, which is the convolution of the input bit, against the states of the convolution encoder, registers. Fundamentally, convolutional codes do not offer more protection against noise than an equivalent block code. In many cases, they generally offer greater simplicity of implementation over a block code of equal power. The encoder is usually a simple circuit which has state memory and some feedback logic, normally XOR gates. The decoder can be implemented in software or firmware. The Viterbi algorithm is the optimum algorithm used to decode convolutional codes. There are simplifications to reduce the computational load. They rely on searching only the most likely paths. Although not optimum, they have generally been found to give good results in the lower noise environments. Convolutional codes are used in voiceband modems (V. V. 1. 7, V. 3. 4) and in GSM mobile phones, as well as satellite and military communication devices. Cryptographical coding. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, and electrical engineering. Applications of cryptography include ATM cards, computer passwords, and electronic commerce. Cryptography prior to the modern age was effectively synonymous with encryption, the conversion of information from a readable state to apparent nonsense. The originator of an encrypted message shared the decoding technique needed to recover the original information only with intended recipients, thereby precluding unwanted persons from doing the same. Since World War I and the advent of the computer, the methods used to carry out cryptology have become increasingly complex and its application more widespread. Modern cryptography is heavily based on mathematical theory and computer science practice; cryptographic algorithms are designed around computational hardness assumptions, making such algorithms hard to break in practice by any adversary. It is theoretically possible to break such a system, but it is infeasible to do so by any known practical means. These schemes are therefore termed computationally secure; theoretical advances, e. There exist information- theoretically secure schemes that provably cannot be broken even with unlimited computing power. Line coding is often used for digital data transport. Line coding consists of representing the digital signal to be transported by an amplitude- and time- discrete signal that is optimally tuned for the specific properties of the physical channel (and of the receiving equipment). The waveform pattern of voltage or current used to represent the 1s and 0s of a digital data on a transmission link is called line encoding. The common types of line encoding are unipolar, polar, bipolar, and Manchester encoding. Other applications of coding theory. A code may be designed so that a phase shift can be easily detected and corrected and that multiple signals can be sent on the same channel. Each phone is assigned a code sequence that is approximately uncorrelated with the codes of other phones. At the receiver, a demodulation process is performed to recover the data. The properties of this class of codes allow many users (with different codes) to use the same radio channel at the same time. To the receiver, the signals of other users will appear to the demodulator only as a low- level noise. In these codes the sender adds redundancy to each message for error checking, usually by adding check bits. If the check bits are not consistent with the rest of the message when it arrives, the receiver will ask the sender to retransmit the message.
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