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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 ...
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]
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 ...
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.
To prove the Elias–Bassalygo bound, start with the following Lemma: Lemma. For C ⊆ [ q ] n {\displaystyle C\subseteq [q]^{n}} and 0 ⩽ e ⩽ n {\displaystyle 0\leqslant e\leqslant n} , there exists a Hamming ball of radius e {\displaystyle e} with at least
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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.