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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] |
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] |
Ball (mathematics)
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defined not only in three-dimensional Euclidean space but also for lower and higher dimensions, and for metric spaces in general. A ''ball'' in dimensions is called a hyperball or -ball and is bounded by a ''hypersphere'' or ()-sphere. Thus, for example, a ball in the Euclidean plane is the same thing as a disk, the area bounded by a circle. In Euclidean 3-space, a ball is taken to be the volume bounded by a 2-dimensional sphere. In a one-dimensional space, a ball is a line segment. In other contexts, such as in Euclidean geometry and informal use, ''sphere'' is sometimes used to mean ''ball''. In the field of topology the closed n-dimensional ball is often denoted as B^n or D^n while the open n-dimensional ball is \operatorname B^n or \ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elias-Bassalygo Bound
The Elias Bassalygo bound is a mathematical limit used in coding theory for error correction during data transmission or communications. Definition Let C be a q-ary code of length n, i.e. a subset of n.Each q-ary block code of length n is a subset of the strings of \mathcal_q^n, where the alphabet set \mathcal_q has q elements. Let R be the ''rate'' of C, \delta the ''relative distance'' and :B_q(y, \rho n) = \left \ be the '' Hamming ball'' of radius \rho n centered at y. Let \text_q(y, \rho n) = , B_q(y, \rho n), be the ''volume'' of the Hamming ball of radius \rho n . It is obvious that the volume of a Hamming Ball is translation-invariant, i.e. indifferent to y. In particular, , B_q(y, \rho n), =, B_q(0, \rho n), . With large enough n, the ''rate'' R and the ''relative distance'' \delta satisfy the Elias-Bassalygo bound: :R \leqslant 1 - H_q ( J_q(\delta))+o(1), where : H_q(x)\equiv_\text -x\log_q \left ( \right)-(1-x)\log_q is the ''q''-ary entropy func ... [...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] |
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] |
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 (Joh ... [...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 si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Binomial Coefficients
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula :\binom nk = \frac, which using factorial notation can be compactly expressed as :\binom = \frac. For example, the fourth power of is :\begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for n=0,1,2,\ldots gives a triangular array called Pascal's triangle, satisfying the recurrence relation :\binom = \binom + \binom. The binomial coefficients occur in many areas of mathematics, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Union (set Theory)
In set theory, the union (denoted by ∪) of a collection of sets is the set of all elements in the collection. It is one of the fundamental operations through which sets can be combined and related to each other. A refers to a union of zero (0) sets and it is by definition equal to the empty set. For explanation of the symbols used in this article, refer to the table of mathematical symbols. Union of two sets The union of two sets ''A'' and ''B'' is the set of elements which are in ''A'', in ''B'', or in both ''A'' and ''B''. In set-builder notation, :A \cup B = \. For example, if ''A'' = and ''B'' = then ''A'' ∪ ''B'' = . A more elaborate example (involving two infinite sets) is: : ''A'' = : ''B'' = : A \cup B = \ As another example, the number 9 is ''not'' contained in the union of the set of prime numbers and the set of even numbers , because 9 is neither prime nor even. Sets cannot have duplicate elements, so the union of the sets and is . Multip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edgar Gilbert
Edgar Nelson Gilbert (July 25, 1923 – June 15, 2013) was an American mathematician and coding theorist, a longtime researcher at Bell Laboratories whose accomplishments include the Gilbert–Varshamov bound in coding theory, the Gilbert–Elliott model of bursty errors in signal transmission, and the Erdős–Rényi model for random graphs. Biography Gilbert was born in 1923 in Woodhaven, New York. He did his undergraduate studies in physics at Queens College, City University of New York, graduating in 1943. He taught mathematics briefly at the University of Illinois at Urbana–Champaign but then moved to the Radiation Laboratory at the Massachusetts Institute of Technology, where he designed radar antennas from 1944 to 1946. He finished a Ph.D. in physics at MIT in 1948, with a dissertation entitled ''Asymptotic Solution of Relaxation Oscillation Problems'' under the supervision of Norman Levinson, and took a job at Bell Laboratories where he remained for the rest of his caree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
