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Justesen Code
In coding theory, Justesen codes form a class of error-correcting codes that have a constant rate, constant relative distance, and a constant alphabet size. Before the Justesen error correction code was discovered, no error correction code was known that had all of these three parameters as a constant. Subsequently, other ECC codes with this property have been discovered, for example expander codes. These codes have important applications in computer science such as in the construction of small-bias sample spaces. Justesen codes are derived as the code concatenation of a Reed–Solomon code and the Wozencraft ensemble. The Reed–Solomon codes used achieve constant rate and constant relative distance at the expense of an alphabet size that is ''linear'' in the message length. The Wozencraft ensemble is a family of codes that achieve constant rate and constant alphabet size, but the relative distance is only constant for most of the codes in the family. The concatenation o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear 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 polynomia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Space
In computational complexity theory, L (also known as LSPACE or DLOGSPACE) is the complexity class containing decision problems that can be solved by a deterministic Turing machine using a logarithmic amount of writable memory space., Definition 8.12, p. 295., p. 177. Formally, the Turing machine has two tapes, one of which encodes the input and can only be read, whereas the other tape has logarithmic size but can be read as well as written. Logarithmic space is sufficient to hold a constant number of pointers into the input and a logarithmic number of boolean flags, and many basic logspace algorithms use the memory in this way. Complete problems and logical characterization Every non-trivial problem in L is complete under log-space reductions, so weaker reductions are required to identify meaningful notions of L-completeness, the most common being first-order reductions. A 2004 result by Omer Reingold shows that USTCON, the problem of whether there exists a path ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Code
In coding theory, a linear code is an error-correcting code for which any linear combination of codewords is also a codeword. Linear codes are traditionally partitioned into block codes and convolutional codes, although turbo codes can be seen as a hybrid of these two types. Linear codes allow for more efficient encoding and decoding algorithms than other codes (cf. syndrome decoding). Linear codes are used in forward error correction and are applied in methods for transmitting symbols (e.g., bits) on a communications channel so that, if errors occur in the communication, some errors can be corrected or detected by the recipient of a message block. The codewords in a linear block code are blocks of symbols that are encoded using more symbols than the original value to be sent. A linear code of length ''n'' transmits blocks containing ''n'' symbols. For example, the ,4,3 Hamming code is a linear binary code which represents 4-bit messages using 7-bit codewords. Two distinct c ... [...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] |
Primitive Element (finite Field)
In field theory, a primitive element of a finite field is a generator of the multiplicative group of the field. In other words, is called a primitive element if it is a primitive th root of unity in ; this means that each non-zero element of can be written as for some integer . If is a prime number, the elements of can be identified with the integers modulo . In this case, a primitive element is also called a primitive root modulo . For example, 2 is a primitive element of the field and , but not of since it generates the cyclic subgroup of order 3; however, 3 is a primitive element of . The minimal polynomial of a primitive element is a primitive polynomial. Properties Number of primitive elements The number of primitive elements in a finite field is , where is Euler's totient function, which counts the number of elements less than or equal to which are relatively prime to . This can be proved by using the theorem that the multiplicative group of a finite fie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension (vector Space)
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say V is if the dimension of V is finite, and if its dimension is infinite. The dimension of the vector space V over the field F can be written as \dim_F(V) or as : F read "dimension of V over F". When F can be inferred from context, \dim(V) is typically written. Examples The vector space \R^3 has \left\ as a standard basis, and therefore \dim_(\R^3) = 3. More generally, \dim_(\R^n) = n, and even more generally, \dim_(F^n) = n for any field F. The complex numbers \Complex are both a real and complex vector space; we have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Code Rate
In telecommunication and information theory, the code rate (or information rateHuffman, W. Cary, and Pless, Vera, ''Fundamentals of Error-Correcting Codes'', Cambridge, 2003.) of a forward error correction code is the proportion of the data-stream that is useful (non- redundant). That is, if the code rate is k/n for every bits of useful information, the coder generates a total of bits of data, of which n-k are redundant. If is the gross bit rate or data signalling rate (inclusive of redundant error coding), the net bit rate (the useful bit rate exclusive of error correction codes) is \leq R \cdot k/n. For example: The code rate of a convolutional code will typically be , , , , , etc., corresponding to one redundant bit inserted after every single, second, third, etc., bit. The code rate of the octet oriented Reed Solomon block code denoted RS(204,188) is 188/204, meaning that redundant octets (or bytes) are added to each block of 188 octets of useful information. A few er ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other results, thi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wozencraft Ensemble
In coding theory, the Wozencraft ensemble is a set of linear codes in which most of codes satisfy the Gilbert-Varshamov bound. It is named after John Wozencraft, who proved its existence. The ensemble is described by , who attributes it to Wozencraft. used the Wozencraft ensemble as the inner codes in his construction of strongly explicit asymptotically good code. Existence theorem :Theorem: Let \varepsilon > 0. For a large enough k, there exists an ensemble of inner codes C_^1,\cdots,C_^N of rate \tfrac, where N = q^k - 1, such that for at least (1 - \varepsilon)N values of i, C_^i has relative distance \geqslant H_q^ \left(\tfrac - \varepsilon \right ). Here relative distance is the ratio of minimum distance to block length. And H_q is the q-ary entropy function defined as follows: :H_q(x) = x\log_q(q-1)-x\log_qx-(1-x)\log_q(1-x). In fact, to show the existence of this set of linear codes, we will specify this ensemble explicitly as follows: for \alpha \in \mathbb_ - \, ... [...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] |
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] |
Concatenated Error Correction Code
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] |
