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| | Even when the BMI is inside the healthy range, other factors could grow a dangers for health issues. One of these factors is excess abdominal fat. Other factors include a family history of health issues, such as diabetes, heart condition, or excellent cholesterol. More information on alternative factors which make losing fat important could be found online.<br><br>Whether you're healthy or not has a excellent deal to do with all the correlation between your weight plus a height plus where the superfluous pounds are accumulating. If you never cognise how to puzzle out a Body Mass Index, you are capable to employ free online [http://safedietplansforwomen.com/bmi-chart bmi chart] calculator to check where you be.<br><br>35.Qualifying for Boston/The Boston Times: Boston is a fantastic, tough race. It is an honor to run it. This is bmi chart men not 1 to be missed in the event you qualify. See some of the posts regarding the Boston Marathon. Check out the Boston Marathon Qualifying Times.<br><br>Another advantage of green leafy vegetables is the fact that they offer phytonutrients, that are nutrients mandatory for sustaining human health by preventing mobile damage, preventing cancer mobile replication and lowering cholesterol levels. They also are a source of vitamins C, E, E plus countless of the B vitamins. Dark green leafy veggies contain small amounts of omega-3 fats.<br><br>Training Programs: A small planning goes a extended means. If possible, try to plan a training to run more usually on softer surfaces like trails, dirt roads, grassy parks, or the track. A few good programs are on the resource page. There are many wise ones out there--find 1 which matches we.<br><br>BMI refuses to measure body fat directly bmi chart women, however, it relates closely to direct measures of body fat. For adults, BMI is interpreted while factors like sex or age are not taken in account.<br><br>Plus, theres a superior chance which if her marriage is inside trouble she could have been deprived in this department for quite some time. Once she finds somebody to take all of which frustration out on, there is no turning back!<br><br>If you don't fall in the normal range, then receive yourself checked with other appropriate techniques to figure out the amount of body fat. This might provide a greater perspective plus help we to achieve or keep the perfect weight. |
| [[File:Braille closeup.jpg|thumb|[[Braille#Contractions|Braille]], a code that uses [[data compression]] extensively to compensate for slower reading times.]]
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| '''Coding theory''' is the study of the properties of [[code]]s and their fitness for a specific application. Codes are used for [[data compression]], [[cryptography]], [[error-correction]] and more recently also for [[network coding]]. Codes are studied by various scientific disciplines—such as [[information theory]], [[electrical engineering]], [[mathematics]], and [[computer science]]—for the purpose of designing efficient and reliable [[data transmission]] methods. This typically involves the removal of redundancy and the correction (or detection) of errors in the transmitted data.
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| There are essentially two aspects to coding theory:
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| # Data compression (or, ''source coding'')
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| # [[Error-correction code|Error correction]] (or ''[[channel coding]]'')
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| These two aspects may be [[Joint source and channel coding|studied in combination]]. Source encoding attempts to compress the data from a source in order to transmit it more efficiently. This practice is found every day on the Internet where the common [[Zip (file format)|Zip data compression]] is used to reduce the network load and make files smaller. The second, channel encoding, 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 channel coding. A typical music CD uses the [[Reed-Solomon error correction#Data storage|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 code]]s.
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| ==History of coding theory==
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| In 1948, [[Claude Shannon]] published "[[A Mathematical Theory of Communication]]", an article in two parts in the July and October issues of the ''Bell System Technical Journal''. 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]].
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| The [[binary Golay code]] was developed in 1949. More specifically, it is an error-correcting code capable of correcting up to three errors in each 24-bit word, and detecting a fourth.
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| [[File:Hamming.jpg|thumb|left|A two-dimensional visualisation<br>of the [[Hamming distance]]]] | |
| [[Richard Hamming]] won the [[Turing Award]] in 1968 for his work at [[Bell Labs]] in numerical methods, automatic coding systems, and error-detecting and error-correcting codes. He invented the concepts known as [[Hamming code]]s, [[Hamming window]]s, [[Hamming numbers]], and [[Hamming distance]].
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| ==Source coding==
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| {{main|Data compression}}
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| The aim of source coding is to take the source data and make it smaller.
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| ===Definition===
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| Data can be seen as a [[random variable]] <math>X:\Omega\rightarrow\mathcal{X}</math>, where <math>x \in \mathcal{X}</math> appears with probability <math>\mathbb{P}[X=x]</math>.
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| Data are encoded by strings (words) over an [[Alphabet (computer science)|alphabet]] <math>\Sigma</math>.
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| A code is a function
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| <math>C:\mathcal{X}\rightarrow\Sigma^*</math> (or <math>\Sigma^+</math> if the empty string is not part of the alphabet).
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| <math>C(x)</math> is the code word associated with <math>x</math>.
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| Length of the code word is written as
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| <math>l(C(x))</math>.
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| Expected length of a code is
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| <math>l(C) = \sum_{x\in\mathcal{X}}l(C(x))\mathbb{P}[X=x]</math>
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| The concatenation of code words <math>C(x_1,...,x_k) = C(x_1)C(x_2)...C(x_k)</math>. | |
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| The code word of the empty string is the empty string itself:
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| <math>C(\epsilon) = \epsilon</math>
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| ===Properties===
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| # <math>C:\mathcal{X}\rightarrow\Sigma^*</math> is [[Variable-length code#Non-singular codes|non-singular]] if [[Injective function|injective]].
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| # <math>C:\mathcal{X}^*\rightarrow\Sigma^*</math> is [[Uniquely decodable code#Uniquely decodable codes|uniquely decodable]] if injective.
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| # <math>C:\mathcal{X}\rightarrow\Sigma^*</math> is [[Variable-length code#Prefix codes|instantaneous]] if <math>C(x_1)</math> is not a prefix of <math>C(x_2)</math> (and vice-versa).
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| ===Principle===
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| [[Entropy (information theory)|Entropy]] of a source is the measure of information. Basically, source codes try to reduce the redundancy present in the source, and represent the source with fewer bits that carry more information.
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| Data compression which explicitly tries to minimize the average length of messages according to a particular assumed probability model is called [[entropy encoding]].
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| Various techniques used by source coding schemes try to achieve the limit of Entropy of the source. ''C''(''x'') ≥ ''H''(''x''), where ''H''(''x'') is entropy of source (bitrate), and ''C''(''x'') is the bitrate after compression. In particular, no source coding scheme can be better than the entropy of the source.
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| ===Example===
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| [[FAX|Facsimile]] transmission uses a simple [[Run-length encoding|run length code]].
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| Source coding removes all data superfluous to the need of the transmitter,
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| decreasing the bandwidth required for transmission.
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| ==Channel coding==
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| {{main|Forward error correction}}
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| The purpose of channel coding theory is to find codes which transmit quickly, contain many valid [[code word]]s and can correct or at least [[error detection|detect]] many errors. 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.{{Citation needed|date=July 2008}} The data is spread out over the disk. Although not a very good code, a simple repeat code can serve as an understandable example. 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. We interleave them. 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 "burst" error of a scratch or a dust spot when this interleaving technique is used.
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| Other codes are more appropriate for different applications. 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.{{Citation needed|date=July 2008}} Cell phones are subject to rapid fading. 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.{{Citation needed|date=July 2008}}
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| ===Linear codes===
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| {{Main|Linear code}}
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| 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.{{Citation needed|date=July 2008}}
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| Algebraic coding theory is basically divided into two major types of codes:{{Citation needed|date=July 2008}}
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| # Linear block codes
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| # Convolutional codes.
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| It analyzes the following three properties of a code – mainly:{{Citation needed|date=July 2008}}
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| *code word length
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| *total number of valid code words
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| *the minimum [[distance]] between two valid code words, using mainly the [[Hamming distance]], sometimes also other distances like the [[Lee distance]].
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| ====Linear block codes====
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| {{Main|Block code}}
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| Linear block codes have the property of [[linearity]], i.e. the sum of any two codewords is also a code word, and they are applied to the source bits in blocks, hence the name linear block codes. There are block codes that are not linear, but it is difficult to prove that a code is a good one without this property.<ref name=terras/>
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| Linear block codes are summarized by their symbol alphabets (e.g., binary or ternary) and parameters (''n'',''m'',''d<sub>min</sub>'')<ref name=blahut/> where
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| # n is the length of the codeword, in symbols,
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| # m is the number of source symbols that will be used for encoding at once,
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| # ''d<sub>min</sub>'' is the minimum hamming distance for the code.
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| {{Merge to |Block code |date=July 2010}}
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| There are many types of linear block codes, such as
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| # [[Cyclic code]]s (e.g., [[Hamming code]]s)
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| # [[Repetition code]]s
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| # [[Parity bit|Parity codes]]
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| # [[Polynomial code]]s (e.g., [[BCH code]]s)
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| # [[Reed-Solomon error correction|Reed–Solomon codes]]
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| # [[Algebraic geometric code]]s
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| # [[Reed–Muller code]]s
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| # [[Hamming bound|Perfect codes]].
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| Block codes are tied to the [[sphere packing]] problem, which has received some attention over the years. 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 (24,12) [[Binary Golay code|Golay code]] used in deep space communications uses 24 dimensions. If used as a binary code (which it usually is) the dimensions refer to the length of the codeword as defined above.
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| 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 "perfect" codes. The only nontrivial and useful perfect codes are the distance-3 Hamming codes with parameters satisfying (2<sup>''r''</sub> – 1, 2<sup>''r''</sub> – 1 – ''r'', 3), and the [23,12,7] binary and [11,6,5] ternary Golay codes.<ref name=terras>{{cite book | title = Fourier Analysis on Finite Groups and Applications |first=Audrey |last=Terras |authorlink=Audrey Terras| publisher = [[Cambridge University Press]] | year = 1999 | isbn = 0-521-45718-1 | url = http://books.google.com/books?id=-B2TA669dJMC&pg=PA195 }}</ref><ref name=blahut>{{cite book |title = Algebraic Codes for Data Transmission |first=Richard E. |last=Blahut | publisher = Cambridge University Press | year = 2003 | isbn = 0-521-55374-1 | url = http://books.google.com/books?id=n0XHMY58tL8C&pg=PA60}}</ref>
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| Another code property is the number of neighbors that a single codeword may have.<ref name=schlegel>
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| {{cite book
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| | title = Trellis and turbo coding
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| | author = Christian Schlegel and Lance Pérez
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| | publisher = Wiley-IEEE
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| | year = 2004
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| | isbn = 978-0-471-22755-7
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| | page = 73
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| | url = http://books.google.com/books?id=9wRCjfGAaEcC&pg=PA73
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| }}</ref>
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| Again, consider pennies as an example. 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.<ref name=schlegel/>
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| Properties of linear block codes are used in many applications. For example, the syndrome-coset uniqueness property of linear block codes is used in trellis shaping,<ref>{{cite journal |first=G.D., Jr. |last=Forney |authorlink=Dave Forney |title=Trellis shaping |journal=IEEE Transactions on Information Theory |volume=38 |issue=2 Pt 2 |pages=281–300 |date=March 1992 |doi=10.1109/18.119687o |url=http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=119687}}</ref> one of the best known [[shaping codes]]. This same property is used in sensor networks for distributed source coding
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| ====Convolutional codes====
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| {{Main|Convolutional code}}
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| The idea behind a convolutional code is to make every codeword symbol be the weighted sum of the various input message symbols. This is like [[convolution]] used in [[linear time invariant|LTI]] systems to find the output of a system, when you know the input and impulse response.
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| 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.
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| 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 [[Decoding methods|decoder]] can be implemented in software or firmware.
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| 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.
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| Convolutional codes are used in voiceband modems (V.32, V.17, V.34) and in GSM mobile phones, as well as satellite and military communication devices.
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| ==Other applications of coding theory==
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| {{misleading|date=August 2012}}
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| Another concern of coding theory is designing codes that help [[synchronization]]. A code may be designed so that a [[Phase (waves)|phase shift]] can be easily detected and corrected and that multiple signals can be sent on the same channel.{{Citation needed|date=July 2008}}
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| Another application of codes, used in some mobile phone systems, is [[code-division multiple access]] (CDMA). Each phone is assigned a code sequence that is approximately uncorrelated with the codes of other phones.{{Citation needed|date=July 2008}} When transmitting, the code word is used to modulate the data bits representing the voice message. 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.{{Citation needed|date=July 2008}}
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| Another general class of codes are the [[automatic repeat-request]] (ARQ) codes. 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. All but the simplest [[wide area network]] protocols use ARQ. Common protocols include [[Synchronous Data Link Control|SDLC]] (IBM), [[Transmission Control Protocol|TCP]] (Internet), [[X.25]] (International) and many others. There is an extensive field of research on this topic because of the problem of matching a rejected packet against a new packet. Is it a new one or is it a retransmission? Typically numbering schemes are used, as in TCP.{{cite web |url= http://tools.ietf.org/html/rfc793 |title= RFC793 |work= RFCs|publisher= [[Internet Engineering Task Force]] (IETF) |date= September 1981}}
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| === Group testing ===
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| [[Group testing]] uses codes in a different way. Consider a large group of items in which a very few are different in a particular way (e.g., defective products or infected test subjects). The idea of group testing is to determine which items are "different" by using as few tests as possible. The origin of the problem has its roots in the [[Second World War]] when the [[United States Army Air Forces]] needed to test its soldiers for [[syphilis]]. It originated from a ground-breaking paper by [[Robert Dorfman]].
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| ===Analog coding===
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| Information is encoded analogously in the [[neural network]]s of [[brain]]s, in [[analog signal processing]], and [[analog electronics]]. Aspects of [http://scholar.google.com/scholar?&q=%22analog+code%22+OR+%22analogue+code%22+OR+%22analogue+coding%22+OR+%22analog+coding%22 analog coding] include [http://scholar.google.com/scholar?q=%22analog+error+correction%22 analog error correction],<ref>{{cite journal |title = Analog Error-Correcting Codes Based on Chaotic Dynamical Systems | id = {{citeseerx|10.1.1.30.4093}} | first1 = Brian | last1 = Chen | first2 = Gregory W. | last2 = Wornell | journal = IEEE Transactions on Communications | volume = 46 | issue = 7 |date=July 1998 |pages=881–890 |doi=10.1109/26.701312 |url=http://allegro.mit.edu/dspg/publications/Journals/pdf/98Chen.pdf |format=PDF}}</ref> [http://scholar.google.com/scholar?q=%22analog+data+compression%22 analog data compression].<ref>{{cite conference | title = On Analog Signature Analysis | id = {{citeseerx|10.1.1.142.5853}} | first1 = Franc Novak Bojan | last1 = Hvala | first2 = Sandi | last2 = Klavžar | booktitle = Proceedings of the conference on Design, automation and test in Europe | year = 1999 | isbn = 1-58113-121-6 }}</ref> [http://scholar.google.com/scholar?hl=en&q=%22analog+encryption%22 analog encryption]<ref>{{cite journal |author=Shujun Li, Chengqing Li, Kwok-Tung Lo, Guanrong Chen |title=Cryptanalyzing an Encryption Scheme Based on Blind Source Separation |journal=IEEE Transactions on Circuits and Systems I |volume=55 |issue=4 |pages=1055–63 |date=April 2008 |doi=10.1109/TCSI.2008.916540 }}</ref>
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| ==Neural coding==
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| [[Neural coding]] is a [[neuroscience]]-related field concerned with how sensory and other information is represented in the [[brain]] by [[neural network|networks]] of [[neurons]]. The main goal of studying neural coding is to characterize the relationship between the [[Stimulus (physiology)|stimulus]] and the individual or ensemble neuronal responses and the relationship among electrical activity of the neurons in the ensemble.<ref name="Brown">{{cite journal |author=Brown EN, Kass RE, Mitra PP |title=Multiple neural spike train data analysis: state-of-the-art and future challenges |journal=Nat. Neurosci. |volume=7 |issue=5 |pages=456–61 |date=May 2004 |pmid=15114358 |doi=10.1038/nn1228 }}</ref> It is thought that neurons can encode both [[Digital data|digital]] and [[analog signal|analog]] information,<ref>{{cite book |first=S.J. |last=Thorpe |chapter=Spike arrival times: A highly efficient coding scheme for neural networks |chapterurl=http://pop.cerco.ups-tlse.fr/fr_vers/documents/thorpe_sj_90_91.pdf |format=PDF |pages=91–94 |editor1-first=R. |editor1-last=Eckmiller |editor2-first=G. |editor2-last=Hartmann |editor3-first=G. |editor3-last=Hauske |title=Parallel processing in neural systems and computers |url=http://books.google.com/books?id=b9gmAAAAMAAJ |accessdate=30 June 2013 |year=1990 |publisher=North-Holland |isbn=978-0-444-88390-2}}</ref> and that neurons follow the principles of information theory and compress information,<ref>{{cite journal |first1=T. |last1=Gedeon |first2=A.E. |last2=Parker |first3=A.G. |last3=Dimitrov |title=Information Distortion and Neural Coding |journal=Canadian Applied Mathematics Quarterly |volume=10 |issue=1 |pages=10 |date=Spring 2002 |url=http://www.math.ualberta.ca/ami/CAMQ/table_of_content/vol_10/10_1c.htm |id={{citeseerx|10.1.1.5.6365}} }}</ref> and detect and correct<ref>{{cite journal |first=M. |last=Stiber |title=Spike timing precision and neural error correction: local behavior |journal=Neural Computation |volume=17 |issue=7 |pages=1577–1601 |date=July 2005 |doi=10.1162/0899766053723069 |url=http://www.mitpressjournals.org/doi/abs/10.1162/0899766053723069 |arxiv=q-bio/0501021}}</ref> errors in the signals that are sent throughout the brain and wider nervous system.
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| ==See also==
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| *[[Coding gain]]
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| *[[Covering code]]
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| *[[Error-correcting code]]
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| *[[Group testing]]
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| *[[Hamming distance]], [[Hamming weight]]
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| *[[Information theory]]
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| *[[Lee distance]]
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| * Spatial coding and [[MIMO]] in [[multiple antenna research]]
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| ** [[Space–time code|Spatial diversity coding]] is spatial coding that transmits replicas of the information signal along different spatial paths, so as to increase the reliability of the data transmission.
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| ** [[Dirty paper coding (DPC)|Spatial interference cancellation coding]]
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| ** [[Spatial multiplexing|Spatial multiplex coding]]
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| *[[Timeline of information theory|Timeline of information theory, data compression, and error correcting codes]]
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| *[[List of algebraic coding theory topics]]
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| *[[Folded Reed–Solomon codes]]
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| *[[ABNNR and AEL codes]]
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| ==Notes==
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| {{Refimprove|date=January 2007}}
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| {{reflist|2}}
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| ==References==
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| *[[Vera Pless]] (1982), ''Introduction to the Theory of Error-Correcting Codes'', John Wiley & Sons, Inc., ISBN 0-471-08684-3.
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| *[[Elwyn R. Berlekamp]] (1984), ''Algebraic Coding Theory'', Aegean Park Press (revised edition), ISBN 0-89412-063-8, ISBN 978-0-89412-063-3.
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| *Randy Yates, ''[http://www.digitalsignallabs.com/tutorial.pdf A Coding Theory Tutorial]''.
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| {{DEFAULTSORT:Coding Theory}}
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| [[Category:Coding theory| ]]
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| [[Category:Error detection and correction]]
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