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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. Varshamov proved ...
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.
These codes attracted interest in the coding theory community because they have the ability to surpass the Gilbert–Varshamov bound; at the time this was discovered, the Gilbert–Varshamov bound had not been broken in the 30 years since its discovery. [6]
Rom Rubenovich Varshamov (Russian Ром Рубенович Варшамов; Born April 9, 1927, in Tbilisi; Died August 24, 1999, in Moscow) was a Soviet Armenian mathematician who worked in Coding theory, especially on error-correcting codes and Number theory.
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I propose to merge Gilbert–Varshamov bound for linear codes into Gilbert–Varshamov bound.I think that the content in the Gilbert–Varshamov bound for linear codes article can easily be explained in the context of Gilbert–Varshamov bound, and the Gilbert–Varshamov bound article is of a reasonable size that the merging of Foo will not cause any problems as far as article size is concerned.
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.