Hamming bound - Revision history

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en>BD2412: /* Perfect codes */repairing Disambiguation pages with links using AWB

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	~~{{Refimprove\|date=December 2009}}~~		I am Beth from Thionville. I am learning to play the Xylophone. Other hobbies are American football.<br><br>Here is my weblog; [http://fitness.makaylaforwomen.com/?p=74%FFsix+guns+Hack%FFnofollow%FFIssac%FFexternal+nofollow%FFCrack+The+Sims+4%FFexternal+nofollow FIFA coin generator]
	~~In [[coding theory]], the '''Gilbert–Varshamov bound''' (due~~ to [[Edgar Gilbert]]<ref>{{citation\|first=E. N.\|last=Gilbert\|authorlink=Edgar Gilbert\|title=A comparison of signalling alphabets\|journal=[[Bell System Technical Journal]]\|volume=31\|year=1952\|pages=504–522}}.</ref> and independently Rom Varshamov<ref>{{citation\|first=R. R.\|last=Varshamov\|title=Estimate of the number of signals in error correcting codes\|journal=Dokl. Acad. Nauk SSSR\|volume=117\|year=1957\|pages=739–741}}.</ref>) is a limit on the parameters of a (not necessarily [[linear code\|linear]]) [[code]]. It is occasionally known as the '''Gilbert–[[Claude Shannon\|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 code. For more about that proof, see: [[GV-linear-code]].

	~~==Statement of the bound==~~
	~~Let~~

	~~:<math>A_q(n,d)</math>~~

	~~denote the maximum possible size of a ''q''-ary code <math>C</math> with length ''n'' and minimum [[Hamming weight]] ''d'' (a ''q''-ary code is a code over~~ the ~~[[field (mathematics)\|field]] <math>\mathbb{F}_q</math> of ''q'' elements).~~

	~~Then:~~

	~~:<math>A_q(n,d) \geq \frac{q^n}{\sum_{j=0}^{d-1} \binom{n}{j}(q-1)^j}~~.~~</math>~~

	~~==Proof==~~
	~~Let <math>C</math> be a code of length <math>n</math> and minimum [[Hamming distance]] <math>d</math> having maximal size:~~

	~~:<math>\|C\|=A_q(n,d).\,</math>~~

	Then for all <math>x\in\mathbb{F}_q^n</math> , there exists at least one codeword <math>c_x \in C</math> such that the Hamming distance <math>d(x,c_x)</math> between <math>x</math> and <math>c_x</math> satisfies

	~~:<math>d(x,c_x)\leq d-1</math>~~

	~~since otherwise we could add ''x'' to the code whilst maintaining the code's minimum Hamming distance ''d'' – a contradiction on the maximality of <math>\|C\|</math>~~.

	~~Hence the whole of~~ <~~math~~>~~\mathbb{F}_q^n~~<~~/math~~> is ~~contained in the [[union (set theory)\|union]] of all [[ball (mathematics)\|balls]] of radius ''d'' −&nbsp~~;~~1 having their [~~[~~ball (mathematics)\|centre]] at some <math>c \in C</math> :~~

	~~:<math>\mathbb{F}_q^n =\cup_{c \in C} B(c,d-1).\, </math>~~

	~~Now each ball has size~~

	~~:<math>~~
	~~\sum_{j=0}^{d-1} \binom{n}{j}(q-1)^j~~
	~~</math>~~

	since we may allow (or [[binomial coefficients\|choose]]) up to <math>d-1</math> of the <math>n</math> components of a codeword to deviate (from the value of the corresponding component of the ball's [[ball (mathematics)\|centre]]) to one of <math>(q-1)</math> possible other values (recall: the code is q-ary: it takes values in <math>\mathbb{F}_q^n</math>). Hence we deduce

	~~:<math>~~
	~~\begin{align}~~
	~~\|\mathbb{F}_q^n\| & = \|\cup_{c \in C} B(c,d-1)\| \\~~
	\\
	~~& \leq \sum_{c \in C} \|B(c,d-1)\| \\~~
	\\
	~~& = \|C\|\sum_{j=0}^{d-1} \binom{n}{j}(q-1)^j \\~~
	\\
	~~\end{align}~~
	~~</math>~~

	~~That is:~~

	:~~<math>~~
	~~A_q(n,d) \geq \frac{q^n}{\sum_{j=0}^{d-1} \binom{n}{j}(q-1)^j}~~
	</~~math>~~

	~~(using the fact: <math>\|\mathbb{F}_q^n\|=q^n<~~/~~math>)~~.

	~~==An improvement in the prime power case==~~
	~~For ''q'' a prime power, one can improve the bound to <math>A_q(n,d)\ge q^k</math> where ''k'' is the greatest integer for which~~

	~~: <math>q^k < \frac{q^n}{\sum_{j=0}^{d-2} \binom{n-1}{j}(q-1)^j}~~.</~~math>~~

	~~==See also~~==
	*[[Singleton bound]]
	*[[Hamming bound]]
	*[[Johnson bound]]
	*[[Plotkin bound]]
	*[[Griesmer bound]]
	*[[Grey–Rankin bound]]
	*[[GV-linear-code\|Gilbert–Varshamov bound for linear code]]
	*[[Elias-Bassalygo bound]]

	~~==References==~~

	~~{{Reflist}}~~

	~~{{DEFAULTSORT:Gilbert-Varshamov Bound}}~~
	~~[[Category:Coding theory]]~~
	~~[[Category:Articles containing proofs]~~]

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2014-01-10T00:21:43Z

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