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Zyablov Bound
In coding theory, the Zyablov bound is a lower bound on the rate r and relative distance \delta that are achievable by concatenated codes. Statement of the bound The bound states that there exists a family of q-ary (concatenated, linear) codes with rate r and relative distance \delta whenever r \leqslant \max\limits_ r' \cdot \left (1 - \right ), where H_qis the q-ary entropy function H_q(x) = x \log_q(q-1) - x \log_q(x) - (1 - x) \log_q(1 - x). Description The bound is obtained by considering the range of parameters that are obtainable by concatenating a "good" outer code C_ with a "good" inner code C_. Specifically, we suppose that the outer code meets the Singleton bound, i.e. it has rate r_and relative distance \delta_ satisfying r_ + \delta_ = 1. Reed Solomon codes are a family of such codes that can be tuned to have ''any'' rate r_ \in (0,1) and relative distance 1 - r_ (albeit over an alphabet as large as the codeword length). We suppose that the inner code meets the G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Concatenated 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, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singleton Bound
In coding theory, the Singleton bound, named after Richard Collom Singleton, is a relatively crude upper bound on the size of an arbitrary block code C with block length n, size M and minimum distance d. It is also known as the Joshibound. proved by and even earlier by . Statement of the bound The minimum distance of a set C of codewords of length n is defined as d = \min_ d(x,y) where d(x,y) is the Hamming distance between x and y. The expression A_(n,d) represents the maximum number of possible codewords in a q-ary block code of length n and minimum distance d. Then the Singleton bound states that A_q(n,d) \leq q^. Proof First observe that the number of q-ary words of length n is q^n, since each letter in such a word may take one of q different values, independently of the remaining letters. Now let C be an arbitrary q-ary block code of minimum distance d. Clearly, all codewords c \in C are distinct. If we puncture the code by deleting the first d-1 letters of each code ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reed Solomon
Reed or Reeds may refer to: Science, technology, biology, and medicine * Reed bird (other) * Reed pen, writing implement in use since ancient times * Reed (plant), one of several tall, grass-like wetland plants of the order Poales * Reed reaction, in chemistry * Reed receiver, an outdated form of multi-channel signal decoding * Reed relay, one or more reed switches controlled by an electromagnet * Reed switch, an electrical switch operated by an applied magnetic field * Reed valve, restricts the flow of fluids to a single direction * Reed (weaving), a comb like tool for beating the weft when weaving * Reed's law, describes the utility of large networks, particularly social networks * Reed–Solomon error correction, a systematic way of building codes that can be used to detect and correct multiple random symbol errors * Reed–Sternberg cell, related to Hodgkin's disease Organizations * Reed (company), offering employment-related services (UK) * Reed and Stem, forme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gilbert–Varshamov Bound
In coding theory, the Gilbert–Varshamov bound (due to Edgar Gilbert and independently Rom Varshamov.) is a limit on the parameters of a (not necessarily linear) code. It is occasionally known as the Gilbert– Shannon–Varshamov bound (or the GSV bound), but the name "Gilbert–Varshamov bound" is by far the most popular. Varshamov proved this bound by using the probabilistic method for linear codes. For more about that proof, see Gilbert–Varshamov bound for linear codes. Statement of the bound Let :A_q(n,d) denote the maximum possible size of a ''q''-ary code C with length ''n'' and minimum Hamming distance ''d'' (a ''q''-ary code is a code over the field \mathbb_q of ''q'' elements). Then: :A_q(n,d) \geqslant \frac. Proof Let C be a code of length n and minimum Hamming distance d having maximal size: :, C, =A_q(n,d). Then for all x\in\mathbb_q^n , there exists at least one codeword c_x \in C such that the Hamming distance d(x,c_x) between x and c_x satisfies :d(x, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reed–Solomon Error Correction
Reed–Solomon codes are a group of error-correcting codes that were introduced by Irving S. Reed and Gustave Solomon in 1960. They have many applications, the most prominent of which include consumer technologies such as MiniDiscs, CDs, DVDs, Blu-ray discs, QR codes, data transmission technologies such as DSL and WiMAX, broadcast systems such as satellite communications, DVB and ATSC, and storage systems such as RAID 6. Reed–Solomon codes operate on a block of data treated as a set of finite-field elements called symbols. Reed–Solomon codes are able to detect and correct multiple symbol errors. By adding = − check symbols to the data, a Reed–Solomon code can detect (but not correct) any combination of up to erroneous symbols, ''or'' locate and correct up to erroneous symbols at unknown locations. As an erasure code, it can correct up to erasures at locations that are known and provided to the algorithm, or it can detect and correct combinations of erro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-polynomial Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expressed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error Detection And Correction
In information theory and coding theory with applications in computer science and telecommunication, error detection and correction (EDAC) or error control are techniques that enable reliable delivery of digital data over unreliable communication channels. Many communication channels are subject to channel noise, and thus errors may be introduced during transmission from the source to a receiver. Error detection techniques allow detecting such errors, while error correction enables reconstruction of the original data in many cases. Definitions ''Error detection'' is the detection of errors caused by noise or other impairments during transmission from the transmitter to the receiver. ''Error correction'' is the detection of errors and reconstruction of the original, error-free data. History In classical antiquity, copyists of the Hebrew Bible were paid for their work according to the number of stichs (lines of verse). As the prose books of the Bible were hardly ever ... [...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] |
Finite Fields
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod when is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number and every positive integer there are fields of order p^k, all of which are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set which is a field; this means that multiplication, addition, subtraction and division (excluding division by zero) are def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
