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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] |
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] |
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] |
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] |
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] |
