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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>&nbsp;, 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;&minus;&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]]

Latest revision as of 18:05, 28 July 2014

Herbal products are an excellent solution to be released from masculine impotence. In other words, these are drugs that cause impotence. Coffee drank excessively causes a debilitating effect upon the sexual organs. Sildenafil citrate, popularly known around the world as Viagra, may be used to take care of impotence problems and pulmonary arterial hypertension. For most people, the major change in marriage comes with the birth of the first child.

One easy solution is Erectasil, a lotion that will make you stand out. The hypnotist would probably create some recommendations that concentrate on this anxiety to discover the invisible problem. In this condition, a man suffers from an inability to develop or maintain an erection sufficient for satisfactory sexual activity. Laboratory studies have shown a significant increase in sexual function in rodents. Levitra takes it easy the effortless muscles of the penis and advances the flow of the blood into the penis, when men are sexually aroused.

There are many different medications out on the market but these medications have a great multiplicity of side effects. But those who were overweight or had a 42-inch waistline were nearly twice as likely to be hit as men with a 32-inch girth. Wifi enabled laptop sperm (update): Do Wi-Fi enabled laptops kill sperm. Your physician is the best person to guide you regarding the treatment of impotence as he knows your health better than anyone else. An anti-aging effect can be achieved from many herbs as lots of them contain certain antioxidants that help with this effect.

Thymoquinone black seed product is used to treatment has been found to alleviate the toxicity of several anticancer drugs by reducing their toxicity to the kidney, heart, liver and central nervous system. It will help your stylist get an idea of what type of cut or color you may have in mind on a future visit. In fact, hops may be part of the reason why heavy beer drinkers often experience hormonal shifts and develop enlarged bellies and breasts. Specifically, steroids are pills, gels, creams or injections that artificially produce hormones to increase sports performance. L-arginine, a well known amino acid is another common cure suggested for reproductive problem.

If the symptoms are more serious and persistent, oral medication is prescribed. Anyone having side effects from this medication can consult their physician about dosing changes. Not all studies point to such benefits, however, and many conventional doctors remain skeptical. Impotence is a common sexual condition that involves a partial, total or an inconsistent inability to achieve or retain an erection that is satisfactory for the purpose of sexual intercourse. Follow the instructions on container for the required dosage.

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