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Repetition Code
In coding theory, the repetition code is one of the most basic error-correcting codes. In order to transmit a message over a noisy channel that may corrupt the transmission in a few places, the idea of the repetition code is to just repeat the message several times. The hope is that the channel corrupts only a minority of these repetitions. This way the receiver will notice that a transmission error occurred since the received data stream is not the repetition of a single message, and moreover, the receiver can recover the original message by looking at the received message in the data stream that occurs most often. Because of the bad error correcting performance coupled with the low code rate (ratio between useful information symbols and actual transmitted symbols), other error correction codes are preferred in most cases. The chief attraction of the repetition code is the ease of implementation. Code parameters In the case of a binary repetition code, there exist two code words ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coding Theory
Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied by various scientific disciplines—such as information theory, electrical engineering, mathematics, linguistics, 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. There are four types of coding: # Data compression (or ''source coding'') # Error control (or ''channel coding'') # Cryptographic coding # Line coding Data compression attempts to remove unwanted redundancy from the data from a source in order to transmit it more efficiently. For example, ZIP data compression makes data files smaller, for purposes such as to reduce Internet traffic. Data compression a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concatenated Error Correction Codes
In coding theory, concatenated codes form a class of error-correcting codes that are derived by combining an inner code and an outer code. They were conceived in 1966 by Dave Forney as a solution to the problem of finding a code that has both exponentially decreasing error probability with increasing block length and polynomial-time decoding complexity. Concatenated codes became widely used in space communications in the 1970s. Background The field of channel coding is concerned with sending a stream of data at the highest possible rate over a given communications channel, and then decoding the original data reliably at the receiver, using encoding and decoding algorithms that are feasible to implement in a given technology. Shannon's channel coding theorem shows that over many common channels there exist channel coding schemes that are able to transmit data reliably at all rates R less than a certain threshold C, called the channel capacity of the given channel. In fact, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Code
In coding theory, block codes are a large and important family of error-correcting codes that encode data in blocks. There is a vast number of examples for block codes, many of which have a wide range of practical applications. The abstract definition of block codes is conceptually useful because it allows coding theorists, mathematicians, and computer scientists to study the limitations of ''all'' block codes in a unified way. Such limitations often take the form of ''bounds'' that relate different parameters of the block code to each other, such as its rate and its ability to detect and correct errors. Examples of block codes are Reed–Solomon codes, Hamming codes, Hadamard codes, Expander codes, Golay codes, and Reed–Muller codes. These examples also belong to the class of linear codes, and hence they are called linear block codes. More particularly, these codes are known as algebraic block codes, or cyclic block codes, because they can be generated using boolean polynomi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
UART
A universal asynchronous receiver-transmitter (UART ) is a computer hardware device for asynchronous serial communication in which the data format and transmission speeds are configurable. It sends data bits one by one, from the least significant to the most significant, framed by start and stop bits so that precise timing is handled by the communication channel. The electric signaling levels are handled by a driver circuit external to the UART. Two common signal levels are RS-232, a 12-volt system, and RS-485, a 5-volt system. Early teletypewriters used current loops. It was one of the earliest computer communication devices, used to attach teletypewriters for an operator console. It was also an early hardware system for the Internet. A UART is usually an individual (or part of an) integrated circuit (IC) used for serial communications over a computer or peripheral device serial port. One or more UART peripherals are commonly integrated in microcontroller chips. Specialised UA ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fountain Code
In coding theory, fountain codes (also known as rateless erasure codes) are a class of erasure codes with the property that a potentially limitless sequence of encoding symbols can be generated from a given set of source symbols such that the original source symbols can ideally be recovered from any subset of the encoding symbols of size equal to or only slightly larger than the number of source symbols. The term ''fountain'' or ''rateless'' refers to the fact that these codes do not exhibit a fixed code rate. A fountain code is optimal if the original ''k'' source symbols can be recovered from any ''k'' successfully received encoding symbols (i.e., excluding those that were erased). Fountain codes are known that have efficient encoding and decoding algorithms and that allow the recovery of the original ''k'' source symbols from any ''k’'' of the encoding symbols with high probability, where ''k’'' is just slightly larger than ''k''. LT codes were the first practical realization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Erasure Channel
In coding theory and information theory, a binary erasure channel (BEC) is a communications channel model. A transmitter sends a bit (a zero or a one), and the receiver either receives the bit correctly, or with some probability P_e receives a message that the bit was not received ("erased") . Definition A binary erasure channel with erasure probability P_e is a channel with binary input, ternary output, and probability of erasure P_e. That is, let X be the transmitted random variable with alphabet \. Let Y be the received variable with alphabet \, where \text is the erasure symbol. Then, the channel is characterized by the conditional probabilities: :\begin \operatorname X = 0 &= 1 - P_e \\ \operatorname X = 1 &= 0 \\ \operatorname X = 0 &= 0 \\ \operatorname X = 1 &= 1 - P_e \\ \operatorname X = 0 &= P_e \\ \operatorname X = 1 &= P_e \end Capacity The channel capacity of a BEC is 1-P_e, attained with a uniform distribution for X (i.e. half of the inputs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Repeat-accumulate Code
In computer science, repeat-accumulate codes (RA codes) are a low complexity class of error-correcting codes. They were devised so that their ensemble weight distributions are easy to derive. RA codes were introduced by Divsalar ''et al.'' In an RA code, an information block of length is repeated times, scrambled by an interleaver of size , and then encoded by a rate 1 accumulator. The accumulator can be viewed as a truncated rate 1 recursive convolutional encoder with transfer function , but Divsalar ''et al.'' prefer to think of it as a block code whose input block and output block are related by the formula and x_i = x_+z_i for i > 1. The encoding time for RA codes is linear and their rate is 1/q. They are nonsystematic. Irregular Repeat Accumulate Codes Irregular Repeat Accumulate (IRA) Codes build on top of the ideas of RA codes. IRA replaces the outer code in RA code with a Low Density Generator Matrix code. IRA codes first repeats information bits different times, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turbo Code
In information theory, turbo codes (originally in French ''Turbocodes'') are a class of high-performance forward error correction (FEC) codes developed around 1990–91, but first published in 1993. They were the first practical codes to closely approach the maximum channel capacity or Shannon limit, a theoretical maximum for the code rate at which reliable communication is still possible given a specific noise level. Turbo codes are used in 3G/ 4G mobile communications (e.g., in UMTS and LTE) and in ( deep space) satellite communications as well as other applications where designers seek to achieve reliable information transfer over bandwidth- or latency-constrained communication links in the presence of data-corrupting noise. Turbo codes compete with low-density parity-check (LDPC) codes, which provide similar performance. The name "turbo code" arose from the feedback loop used during normal turbo code decoding, which was analogized to the exhaust feedback used for engine tur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error-correcting Code
In computing, telecommunication, information theory, and coding theory, an error correction code, sometimes error correcting code, (ECC) is used for controlling errors in data over unreliable or noisy communication channels. The central idea is the sender encodes the message with redundant information in the form of an ECC. The redundancy allows the receiver to detect a limited number of errors that may occur anywhere in the message, and often to correct these errors without retransmission. The American mathematician Richard Hamming pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming (7,4) code. ECC contrasts with error detection in that errors that are encountered can be corrected, not simply detected. The advantage is that a system using ECC does not require a reverse channel to request retransmission of data when an error occurs. The downside is that there is a fixed overhead that is added to the message, thereby requiring a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Majority Logic Decoding
In error detection and correction, majority logic decoding is a method to decode repetition codes, based on the assumption that the largest number of occurrences of a symbol was the transmitted symbol. Theory In a binary alphabet made of 0,1, if a (n,1) repetition code is used, then each input bit is mapped to the code word as a string of n-replicated input bits. Generally n=2t + 1, an odd number. The repetition codes can detect up to /2/math> transmission errors. Decoding errors occur when more than these transmission errors occur. Thus, assuming bit-transmission errors are independent, the probability of error for a repetition code is given by P_e = \sum_^ \epsilon^ (1-\epsilon)^{(n-k)}, where \epsilon is the error over the transmission channel. Algorithm Assumption: the code word is (n,1), where n=2t+1, an odd number. * Calculate the d_H Hamming weight of the repetition code. * if d_H \le t , decode code word to be all 0's * if d_H \ge t+1 , decode code word to be all 1's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
