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Hamming Space
In statistics and coding theory, a Hamming space is usually the set of all 2^N binary strings of length ''N'', where different binary strings are considered to be ''adjacent'' when they differ only in one position. The total distance between any two binary strings is then the total number of positions at which the corresponding bits are different, called the Hamming distance. Hamming spaces are named after American mathematician Richard Hamming, who introduced the concept in 1950. They are used in the theory of coding signals and transmission. More generally, a Hamming space can be defined over any alphabet (set) ''Q'' as the set of words of a fixed length ''N'' with letters from ''Q''.Cohen et al., ''Covering Codes'', p. 15 If ''Q'' is a finite field, then a Hamming space over ''Q'' is an ''N''-dimensional vector space over ''Q''. In the typical, binary case, the field is thus GF(2) (also denoted by Z2). In coding theory, if ''Q'' has ''q'' elements, then any subset ''C'' (u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamming Distance 3 Bit Binary
Hamming may refer to: * Richard Hamming (1915–1998), American mathematician * Hamming(7,4), in coding theory, a linear error-correcting code * Overacting, or acting in an exaggerated way See also * Hamming code, error correction in telecommunication * Hamming distance, a way of defining how different two sequences are * Hamming weight, the number of non-zero elements in a sequence * Hamming window, a mathematical function used in signal processing * Hammond (other) * Ham (other) {{disambiguation, surname ... [...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 Code word (communication), 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 [7,4,3] Hamming code is a linear binary code which represents 4-bit messages using 7-bit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gray Isometry
The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit). For example, the representation of the decimal value "1" in binary would normally be "", and "2" would be "". In Gray code, these values are represented as "" and "". That way, incrementing a value from 1 to 2 requires only one bit to change, instead of two. Gray codes are widely used to prevent spurious output from electromechanical switches and to facilitate error correction in digital communications such as digital terrestrial television and some cable TV systems. The use of Gray code in these devices helps simplify logic operations and reduce errors in practice. Function Many devices indicate position by closing and opening switches. If that device uses natural binary codes, positions 3 and 4 are next to each other but all three bits of the binary representa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lee Distance
In coding theory, the Lee distance is a distance between two strings x_1 x_2 \dots x_n and y_1 y_2 \dots y_n of equal length ''n'' over the ''q''-ary alphabet of size . It is a metric defined as \sum_^n \min(, x_i - y_i, ,\, q - , x_i - y_i, ). If or the Lee distance coincides with the Hamming distance, because both distances are 0 for two single equal symbols and 1 for two single non-equal symbols. For this is not the case anymore; the Lee distance between single letters can become bigger than 1. However, there exists a Gray isometry (weight-preserving bijection) between \mathbb_4 with the Lee weight and \mathbb_2^2 with the Hamming weight. Considering the alphabet as the additive group Z''q'', the Lee distance between two single letters x and y is the length of shortest path in the Cayley graph (which is circular since the group is cyclic) between them. More generally, the Lee distance between two strings of length is the length of the shortest path between them in the Ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Submodule
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic operations for integers, other than the usual ones from elementary arithmetic, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar example of modular arithmetic is the hour hand on a 12-hour clock. If the hour hand points to 7 now, then 8 hours later it will point to 3. Ordinary addition would result in , but 15 reads as 3 on the clock face. This is because the hour hand makes one rotation every 12 hours and the hour number starts over when the hour hand passes 12. We say that 15 is ''congruent'' to 3 modulo 12, written 15 ≡ 3 (mod 12), so that 7 + 8 ≡ 3 (mod 12). Similarly, if one starts at 12 and waits 8 hours, the hour hand will be at 8. If one instead waited twice as long, 16 hours, the hour hand would be on 4. This ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Ring
In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements. Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite group, but the concept of finite rings in their own right has a more recent history. Although rings have more structure than groups do, the theory of finite rings is simpler than that of finite groups. For instance, the classification of finite simple groups was one of the major breakthroughs of 20th century mathematics, its proof spanning thousands of journal pages. On the other hand, it has been known since 1907 that any finite simple ring is isomorphic to the ring \mathrm_n(\mathbb_q) – the ''n''-by-''n'' matrices over a finite field of order ''q'' (as a consequence of Wedderburn's theorems, described below). The number of rings with ''m'' elements, for ''m'' a natural number, is listed under in the On-Line Encyclopedia of Integer Seq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error Detection And Correction
In information theory and coding theory with applications in computer science and telecommunications, 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 w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric (mathematics)
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are a general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Variable-length Code
In coding theory, a variable-length code is a code which maps source symbols to a ''variable'' number of bits. The equivalent concept in computer science is '' bit string''. Variable-length codes can allow sources to be compressed and decompressed with ''zero'' error (lossless data compression) and still be read back symbol by symbol. With the right coding strategy, an independent and identically-distributed source may be compressed almost arbitrarily close to its entropy. This is in contrast to fixed-length coding methods, for which data compression is only possible for large blocks of data, and any compression beyond the logarithm of the total number of possibilities comes with a finite (though perhaps arbitrarily small) probability of failure. Some examples of well-known variable-length coding strategies are Huffman coding, Lempel–Ziv coding, arithmetic coding, and context-adaptive variable-length coding. Codes and their extensions The extension of a code is the m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
