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Hexacode
In coding theory, the hexacode is a length 6 linear code of dimension 3 over the Galois field GF(4)=\ of 4 elements defined by :H=\. It is a 3-dimensional subspace of the vector space of dimension 6 over GF(4). Then H contains 45 codewords of weight 4, 18 codewords of weight 6 and the zero word. The full automorphism group of the hexacode is 3.S_6. The hexacode can be used to describe the Miracle Octad Generator In mathematics, the Miracle Octad Generator, or MOG, is a mathematical tool introduced by Rob T. Curtis for manipulating the Mathieu groups, binary Golay code and Leech lattice. Description The Miracle Octad Generator is a 4x6 array of combinatio ... of R. T. Curtis. References *{{cite book , first = John H. , last = Conway , authorlink = John Horton Conway , author2=Sloane, Neil J. A. , authorlink2=Neil Sloane , year = 1998 , title = Sphere Packings, Lattices and Groups , url = https://archive.org/details/spherepackingsla0000conw_b8u0 , url-access = registrati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Miracle Octad Generator
In mathematics, the Miracle Octad Generator, or MOG, is a mathematical tool introduced by Rob T. Curtis for manipulating the Mathieu groups, binary Golay code and Leech lattice. Description The Miracle Octad Generator is a 4x6 array of combinations describing any point in 24-dimensional space. It preserves all of the symmetries and maximal subgroups of the Mathieu group M24, namely the monad group, duad group, triad group, octad group, octern group, sextet group, trio group and duum group. It can therefore be used to study all of these symmetries. Golay code Another use for the Miracle Octad Generator is to quickly verify codewords of the binary Golay code. Each element of the Miracle Octad Generator can store either a '1' or a '0', usually displayed as an asterisk and blank space, respectively. Each column and the top row have a property known as the ''count'', which is the number of asterisks in that particular line. One of the criteria for a set of 24 coordinates to be a codewo ... [...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] |
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] |
Galois Field
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] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Codewords
In communication, a code word is an element of a standardized code or protocol. Each code word is assembled in accordance with the specific rules of the code and assigned a unique meaning. Code words are typically used for reasons of reliability, clarity, brevity, or secrecy. See also * Code word (figure of speech) * Coded set * Commercial code (communications) * Compartmentalization (information security) * Duress code * Error correction and detection * Marine VHF radio * Password * Safeword In BDSM, a safeword is a code word, series of code words or other signal used by a person to communicate their physical or emotional state, typically when approaching, or crossing, a physical, emotional, or moral boundary. Some safewords are used ... * Spelling alphabet References * * *UNHCR Procedure for Radio Communication External links UNHCR Procedure for Radio Communication Data transmission Cryptography {{crypto-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weight
In science and engineering, the weight of an object is the force acting on the object due to gravity. Some standard textbooks define weight as a Euclidean vector, vector quantity, the gravitational force acting on the object. Others define weight as a scalar quantity, the magnitude of the gravitational force. Yet others define it as the magnitude of the reaction (physics), reaction force exerted on a body by mechanisms that counteract the effects of gravity: the weight is the quantity that is measured by, for example, a spring scale. Thus, in a state of free fall, the weight would be zero. In this sense of weight, terrestrial objects can be weightless: ignoring Drag (physics), air resistance, the famous apple falling from the tree, on its way to meet the ground near Isaac Newton, would be weightless. The unit of measurement for weight is that of force, which in the International System of Units (SI) is the newton (unit), newton. For example, an object with a mass of one kilogram ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphism Group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the group of invertible linear transformations from ''X'' to itself (the general linear group of ''X''). If instead ''X'' is a group, then its automorphism group \operatorname(X) is the group consisting of all group automorphisms of ''X''. Especially in geometric contexts, an automorphism group is also called a symmetry group. A subgroup of an automorphism group is sometimes called a transformation group. Automorphism groups are studied in a general way in the field of category theory. Examples If ''X'' is a set with no additional structure, then any bijection from ''X'' to itself is an automorphism, and hence the automorphism group of ''X'' in this case is precisely the symmetric group of ''X''. If the set ''X'' has additional struct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
