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Singleton Bound
In coding theory, the Singleton bound, named after Richard Collom Singleton, is a relatively crude upper bound on the size of an arbitrary block code C with block length n, size M and minimum distance d. It is also known as the Joshibound. proved by and even earlier by . Statement of the bound The minimum distance of a set C of codewords of length n is defined as d = \min_ d(x,y) where d(x,y) is the Hamming distance between x and y. The expression A_(n,d) represents the maximum number of possible codewords in a q-ary block code of length n and minimum distance d. Then the Singleton bound states that A_q(n,d) \leq q^. Proof First observe that the number of q-ary words of length n is q^n, since each letter in such a word may take one of q different values, independently of the remaining letters. Now let C be an arbitrary q-ary block code of minimum distance d. Clearly, all codewords c \in C are distinct. If we puncture the code by deleting the first d-1 letters of each code ... [...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 and er ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called " points at infinity") to Euclidean points, and vice-versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because a ... [...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 and er ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge
Cambridge ( ) is a College town, university city and the county town in Cambridgeshire, England. It is located on the River Cam approximately north of London. As of the 2021 United Kingdom census, the population of Cambridge was 145,700. Cambridge became an important trading centre during the Roman and Viking ages, and there is archaeological evidence of settlement in the area as early as the Bronze Age. The first Town charter#Municipal charters, town charters were granted in the 12th century, although modern city status was not officially conferred until 1951. The city is most famous as the home of the University of Cambridge, which was founded in 1209 and consistently ranks among the best universities in the world. The buildings of the university include King's College Chapel, Cambridge, King's College Chapel, Cavendish Laboratory, and the Cambridge University Library, one of the largest legal deposit libraries in the world. The city's skyline is dominated by several Colleg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stamm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Griesmer Bound
In the mathematics of coding theory, the Griesmer bound, named after James Hugo Griesmer, is a bound on the length of linear binary codes of dimension ''k'' and minimum distance ''d''. There is also a very similar version for non-binary codes. Statement of the bound For a binary linear code, the Griesmer bound is: : n\geqslant \sum_^ \left\lceil\frac\right\rceil. Proof Let N(k,d) denote the minimum length of a binary code of dimension ''k'' and distance ''d''. Let ''C'' be such a code. We want to show that : N(k,d)\geqslant \sum_^ \left\lceil\frac\right\rceil. Let ''G'' be a generator matrix of ''C''. We can always suppose that the first row of ''G'' is of the form ''r'' = (1, ..., 1, 0, ..., 0) with weight ''d''. : G= \begin 1 & \dots & 1 & 0 & \dots & 0 \\ \ast & \ast & \ast & & G' & \\ \end The matrix G' generates a code C', which is called the residual code of C. C' obviously has dimension k'=k-1 and length n'=N(k,d)-d. C' has a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johnson Bound
In applied mathematics, the Johnson bound (named after Selmer Martin Johnson) is a limit on the size of error-correcting codes, as used in coding theory for data transmission or communications. Definition Let C be a ''q''-ary code of length n, i.e. a subset of \mathbb_q^n. Let d be the minimum distance of C, i.e. :d = \min_ d(x,y), where d(x,y) is the Hamming distance between x and y. Let C_q(n,d) be the set of all ''q''-ary codes with length n and minimum distance d and let C_q(n,d,w) denote the set of codes in C_q(n,d) such that every element has exactly w nonzero entries. Denote by , C, the number of elements in C. Then, we define A_q(n,d) to be the largest size of a code with length n and minimum distance d: : A_q(n,d) = \max_ , C, . Similarly, we define A_q(n,d,w) to be the largest size of a code in C_q(n,d,w): : A_q(n,d,w) = \max_ , C, . Theorem 1 (Johnson bound for A_q(n,d)): If d=2t+1, : A_q(n,d) \leq \frac. If d=2t+2, : A_q(n,d) \leq \frac. Theorem 2 (J ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamming Bound
In mathematics and computer science, in the field of coding theory, the Hamming bound is a limit on the parameters of an arbitrary block code: it is also known as the sphere-packing bound or the volume bound from an interpretation in terms of packing balls in the Hamming metric into the space of all possible words. It gives an important limitation on the efficiency with which any error-correcting code can utilize the space in which its code words are embedded. A code that attains the Hamming bound is said to be a perfect code. Background on error-correcting codes An original message and an encoded version are both composed in an alphabet of ''q'' letters. Each code word contains ''n'' letters. The original message (of length ''m'') is shorter than ''n'' letters. The message is converted into an ''n''-letter codeword by an encoding algorithm, transmitted over a noisy channel, and finally decoded by the receiver. The decoding process interprets a garbled codeword, referred to as s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plotkin Bound
In the mathematics of coding theory, the Plotkin bound, named after Morris Plotkin, is a limit (or bound) on the maximum possible number of codewords in binary codes of given length ''n'' and given minimum distance ''d''. Statement of the bound A code is considered "binary" if the codewords use symbols from the binary alphabet \. In particular, if all codewords have a fixed length ''n'', then the binary code has length ''n''. Equivalently, in this case the codewords can be considered elements of vector space \mathbb_2^n over the finite field \mathbb_2. Let d be the minimum distance of C, i.e. :d = \min_ d(x,y) where d(x,y) is the Hamming distance between x and y. The expression A_(n,d) represents the maximum number of possible codewords in a binary code of length n and minimum distance d. The Plotkin bound places a limit on this expression. Theorem (Plotkin bound): i) If d is even and 2d > n , then : A_(n,d) \leq 2 \left\lfloor\frac\right\rfloor. ii) If d is odd and ... [...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] |
Arc (projective Geometry)
An (''simple'') arc in finite projective geometry is a set of points which satisfies, in an intuitive way, a feature of ''curved'' figures in continuous geometries. Loosely speaking, they are sets of points that are far from "line-like" in a plane or far from "plane-like" in a three-dimensional space. In this finite setting it is typical to include the number of points in the set in the name, so these simple arcs are called -arcs. An important generalization of the -arc concept, also referred to as arcs in the literature, are the ()-arcs. -arcs in a projective plane In a finite projective plane (not necessarily Desarguesian) a set of points such that no three points of are collinear (on a line) is called a . If the plane has order then , however the maximum value of can only be achieved if is even. In a plane of order , a -arc is called an oval and, if is even, a -arc is called a hyperoval. Every conic in the Desarguesian projective plane PG(2,), i.e., the set of zero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
