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Alternant Code
In coding theory, alternant codes form a class of parameterised error-correcting codes which generalise the BCH codes. Definition An ''alternant code'' over GF(''q'') of length ''n'' is defined by a parity check matrix ''H'' of alternant form ''H''''i'',''j'' = αji''y''''i'', where the α''j'' are distinct elements of the extension GF(''q''''m''), the ''y''''i'' are further non-zero parameters again in the extension GF(''q''''m'') and the indices range as ''i'' from 0 to δ − 1, ''j'' from 1 to ''n''. Properties The parameters of this alternant code are length ''n'', dimension ≥ ''n'' − ''m''δ and minimum distance ≥ δ + 1. There exist long alternant codes which meet the Gilbert–Varshamov bound. The class of alternant codes includes * BCH code In coding theory, the Bose–Chaudhuri–Hocquenghem codes (BCH codes) form a class of cyclic error-correcting codes that are constructed using polynomials over a finite field ... [...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] |
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] |
BCH Code
In coding theory, the Bose–Chaudhuri–Hocquenghem codes (BCH codes) form a class of cyclic error-correcting codes that are constructed using polynomials over a finite field (also called ''Galois field''). BCH codes were invented in 1959 by French mathematician Alexis Hocquenghem, and independently in 1960 by Raj Chandra Bose and D.K. Ray-Chaudhuri. The name ''Bose–Chaudhuri–Hocquenghem'' (and the acronym ''BCH'') arises from the initials of the inventors' surnames (mistakenly, in the case of Ray-Chaudhuri). One of the key features of BCH codes is that during code design, there is a precise control over the number of symbol errors correctable by the code. In particular, it is possible to design binary BCH codes that can correct multiple bit errors. Another advantage of BCH codes is the ease with which they can be decoded, namely, via an algebraic method known as syndrome decoding. This simplifies the design of the decoder for these codes, using small low-pow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alternant Matrix
In linear algebra, an alternant matrix is a matrix formed by applying a finite list of functions pointwise to a fixed column of inputs. An alternant determinant is the determinant of a square alternant matrix. Generally, if f_1, f_2, \dots, f_n are functions from a set X to a field F, and \in X, then the alternant matrix has size m \times n and is defined by :M=\begin f_1(\alpha_1) & f_2(\alpha_1) & \cdots & f_n(\alpha_1)\\ f_1(\alpha_2) & f_2(\alpha_2) & \cdots & f_n(\alpha_2)\\ f_1(\alpha_3) & f_2(\alpha_3) & \cdots & f_n(\alpha_3)\\ \vdots & \vdots & \ddots &\vdots \\ f_1(\alpha_m) & f_2(\alpha_m) & \cdots & f_n(\alpha_m)\\ \end or, more compactly, M_ = f_j(\alpha_i). (Some authors use the transpose of the above matrix.) Examples of alternant matrices include Vandermonde matrices, for which f_j(\alpha)=\alpha^, and Moore matrices, for which f_j(\alpha)=\alpha^. Properties * The alternant can be used to check the linear independence of the functions f_1, f_2, \dots, f_n in ... [...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] |
Binary Goppa Code
In mathematics and computer science, the binary Goppa code is an error-correcting code that belongs to the class of general Goppa codes originally described by Valerii Denisovich Goppa, but the binary structure gives it several mathematical advantages over non-binary variants, also providing a better fit for common usage in computers and telecommunication. Binary Goppa codes have interesting properties suitable for cryptography in McEliece-like cryptosystems and similar setups. Construction and properties A binary Goppa code is defined by a polynomial g(x) of degree t over a finite field GF(2^m) with no repeated roots, and a sequence L_1, ..., L_n of n distinct elements from GF(2^m) that are not roots of g. Codewords belong to the kernel of the syndrome function, forming a subspace of \^n: : \Gamma(g,L)=\left\ The code defined by a tuple (g,L) has dimension at least n-mt and distance at least 2t+1, thus it can encode messages of length at least n-mt using codewords of size n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Srivastava Code
In coding theory, Srivastava codes, formulated by Professor J. N. Srivastava, form a class of parameterised error-correcting codes which are a special case of alternant code In coding theory, alternant codes form a class of parameterised error-correcting codes which generalise the BCH codes. Definition An ''alternant code'' over GF(''q'') of length ''n'' is defined by a parity check matrix ''H'' of alternant form ''H ...s. Definition The original ''Srivastava code'' over GF(''q'') of length ''n'' is defined by a parity check matrix ''H'' of alternant form :\begin \frac & \cdots & \frac \\ \vdots & \ddots & \vdots \\ \frac & \cdots & \frac \\ \end where the α''i'' and ''z''''i'' are elements of GF(''q''''m'') Properties The parameters of this code are length ''n'', dimension ≥ ''n'' − ''m''s and minimum distance ≥ s + 1. References * Error detection and correction Finite fields Coding theory {{crypto-stub ... [...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] |
