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In coding theory, the Gilbert–Varshamov bound (due to Edgar Gilbert [1] and independently Rom Varshamov [2]) 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.
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. This may be translated into a statement about the maximum rate of a code with given length ...
An Improvement is done to the Gilbert-Varshamov bound already discussed above. Using the connection between permutation codes and independent sets in certain graphs one can improve the Gilbert–Varshamov bound asymptotically by a factor log ( n ) {\displaystyle \log(n)} , when the code length goes to infinity.
The Hermitian curve is given by the equation + = + considered over the field . [2] This curve is of particular importance because it meets the Hasse–Weil bound with equality, and thus has the maximal number of affine points over F q 2 {\displaystyle \mathbb {F} _{q^{2}}} . [ 12 ]
Edgar Nelson Gilbert (July 25, 1923 – June 15, 2013) was an American mathematician and coding theorist, a longtime researcher at Bell Laboratories.His accomplishments include the Gilbert–Varshamov bound in coding theory, the Gilbert–Elliott model of bursty errors in signal transmission, the Erdős–Rényi–Gilbert model for random graphs, the Gilbert disk model of random geometric ...
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.
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 with block length , size and minimum distance . It is also known as the Joshibound [ 1 ] proved by Joshi (1958) and even earlier by Komamiya (1953) .
We suppose that the inner code meets the Gilbert–Varshamov bound, i.e. it has rate and relative distance satisfying + (). Random linear codes are known to satisfy this property with high probability, and an explicit linear code satisfying the property can be found by brute-force search (which requires time polynomial in the size of the ...