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Gilbert–Varshamov Bound For Linear Codes
The Gilbert–Varshamov bound for linear codes is related to the general Gilbert–Varshamov bound, which gives a lower bound on the maximal number of elements in an error-correcting code of a given block length and minimum Hamming weight over a field \mathbb_q. This may be translated into a statement about the maximum rate of a code with given length and minimum distance. The Gilbert–Varshamov bound for linear codes asserts the existence of ''q''-ary linear codes for any relative minimum distance less than the given bound that simultaneously have high rate. The existence proof uses the probabilistic method, and thus is not constructive. The Gilbert–Varshamov bound is the best known in terms of relative distance for codes over alphabets of size less than 49. For larger alphabets, algebraic geometry codes sometimes achieve an asymptotically better rate vs. distance tradeoff than is given by the Gilbert–Varshamov bound. Gilbert–Varshamov bound theorem :Theorem: Let q \g ... [...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 bound on the size 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 Recall that a code has a minimum distance d if any two elements in the code are at least a distance d apart. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamming Distance
In information theory, the Hamming distance between two String (computer science), strings or vectors of equal length is the number of positions at which the corresponding symbols are different. In other words, it measures the minimum number of ''substitutions'' required to change one string into the other, or equivalently, the minimum number of ''errors'' that could have transformed one string into the other. In a more general context, the Hamming distance is one of several string metrics for measuring the edit distance between two sequences. It is named after the American mathematician Richard Hamming. A major application is in coding theory, more specifically to block codes, in which the equal-length strings are Vector space, vectors over a finite field. Definition The Hamming distance between two equal-length strings of symbols is the number of positions at which the corresponding symbols are different. Examples The symbols may be letters, bits, or decimal digits, am ... [...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 Sphere packing, packing balls in the Hamming metric into the Space (mathematics), 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 word (communication), 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 (communication), 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 communication channel, ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Las Vegas Algorithm
In computing, a Las Vegas algorithm is a randomized algorithm that always gives Correctness (computer science), correct results; that is, it always produces the correct result or it informs about the failure. However, the runtime of a Las Vegas algorithm differs depending on the input. The usual definition of a Las Vegas algorithm includes the restriction that the ''expected'' runtime be finite, where the expectation is carried out over the space of random information, or entropy, used in the algorithm. An alternative definition requires that a Las Vegas algorithm always terminates (is Effective method, effective), but may output a Partial function#Bottom element, symbol not part of the solution space to indicate failure in finding a solution. The nature of Las Vegas algorithms makes them suitable in situations where the number of possible solutions is limited, and where verifying the correctness of a candidate solution is relatively easy while finding a solution is complex. Systema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamming Ball
In combinatorics, a Hamming ball is a metric ball for Hamming distance. The Hamming ball of radius r centered at a String (computer science), string x over some Alphabet (formal languages), alphabet (often the alphabet ) is the set of all strings of the same length that differ from x in at most r positions. This may be denoted using the standard notation for metric balls, B(s,r). For an alphabet X and a string x, the Hamming ball is a subset of the Hamming space X^ of strings of the same length as x, and it is a proper subset whenever r<, x, . The name ''Hamming ball'' comes from coding theory, where error correction codes can be defined as having disjoint Hamming balls around their codewords, and covering codes can be defined as having Hamming balls around the codeword whose union is the whole Hamming space. Some Local search (constraint satisfaction), local search algorithms for SAT solvers such as WalkSAT operate by using random guessing or covering codes to find a Ham ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
