Gilbert–Varshamov bound for linear codes: Difference between revisions

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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 correction code|error-correcting code]] of a given block length and minimum [[Hamming weight]] over a [[field (mathematics)|field]] <math>\mathbb{F}_q</math>. 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 code|linear codes]] asserts the existence of ''q''-ary linear codes for any rate and relative minimum distance less than the given bounds. The existence proof uses [[probabilistic method]], and thus is not constructive.

In [[coding theory]], the bound of parameters such as rate ''R'', relative distance, [[block length]], etc. is usually concerned. Here [[Gilbert–Varshamov bound|Gilbert–Varshamov bound theorem]] claims the lower bound of the rate of the general code. Gilbert–Varshamov bound is the best in terms of relative distance for codes over alphabets of size less than 49.{{citation needed|date=May 2013}}

The Gilbert–Varshamov bound is the best in terms of relative distance for codes over alphabets of size less than 49.{{citation needed|date=May 2013}}

==Gilbert–Varshamov bound theorem==