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| A '''locally testable code''' is a type of [[error-correcting code]] for which it can be determined if a [[String (computer science)|string]] is a [[Code word|word]] in that code by looking at a small (frequently constant) number of bits of the string. In some situations, it is useful to know if the data is corrupted without decoding all of it so that appropriate action can be taken in response. For example, in communication, if the receiver encounters a corrupted code, it can request the data be re-sent, which could increase the accuracy of said data. Similarly, in data storage, these codes can allow for damaged data to be recovered and rewritten properly.
| | Today, there are several other types of web development and blogging software available to design and host your website blogs online and that too in minutes, if not hours. What I advise you do next is save the backup data file to a remote place like a CD-ROM, external disk drive if you have one or a provider such as Dropbox. One really cool features about this amazing and free wp plugin is that the code it generates is completely portable. Donor oocytes and menopausal pregnancy: Oocyte donation to women of advanced reproductive age: pregnancy results and obstetrical outcomes in patients 45 years and older. provided by Word - Press Automatic Upgrade, so whenever you need to update the new version does not, it automatically creates no webmaster. <br><br>You just download ready made templates to a separate directory and then choose a favorite one in the admin panel. When you write a new post, you'll see a small bar that goes across the text input area. A Wordpress plugin is a software that you can install into your Wordpress site. They provide many such popular products which you can buy for your baby. Now a days it has since evolved into a fully capable CMS platform which make it, the best platform in the world for performing online business. <br><br>The entrepreneurs can easily captivate their readers by using these versatile themes. As of now, Pin2Press is getting ready to hit the market. I've applied numerous Search engine optimization-ready Word - Press themes and I can say from knowledge that I consider the Genesis Search engine marketing panel one particular of the simplest to use. Newer programs allow website owners and internet marketers to automatically and dynamically change words in their content to match the keywords entered by their web visitors in their search queries'a feat that they cannot easily achieve with older software. Premium vs Customised Word - Press Themes - Premium themes are a lot like customised themes but without the customised price and without the wait. <br><br>Whether your Word - Press themes is premium or not, but nowadays every theme is designed with widget-ready. I have compiled a few tips on how you can start a food blog and hopefully the following information and tips can help you to get started on your food blogging creative journey. However, you may not be able to find a theme that is in sync with your business. Fast Content Update - It's easy to edit or add posts with free Wordpress websites. In case you beloved this informative article along with you want to get guidance regarding [http://s.do-dance.com/wordpressbackup914058 wordpress backup] i implore you to pay a visit to our page. Fortunately, Word - Press Customization Service is available these days, right from custom theme design, to plugin customization and modifying your website, you can take any bespoke service for your Word - Press development project. <br><br>A sitemap is useful for enabling web spiders and also on rare occasions clients, too, to more easily and navigate your website. I don't want that for my visitors and I'm quite sure they don't either. Useful Plugins Uber - Menu Top Megamenu Now it is the time of sticky Top navbar. You should stay away from plugins that are full of flaws and bugs. You can check out the statistics of page of views for your web pages using free tools that are available on the internet. |
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| In contrast, [[locally decodable code]]s use a small number of bits of the codeword to [[Probabilistic Turing Machine|probabilistically]] recover the original information. The fraction of errors determines how likely it is that the decoder correctly recovers the original bit; however, not all locally decodable codes are locally testable.<ref name=decNotTest>{{cite web|url=http://eccc.hpi-web.de/report/2010/130/revision/1/download/|title=Locally Testable vs. Locally Decodable Codes|author=Tali Kaufman, Michael Viderman}}</ref>
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| Clearly, any valid codeword should be accepted as a codeword, but strings that are not codewords could be only one bit off, which would require many (certainly more than a constant number) probes. To account for this, testing failure is only defined if the string is off by at least a set fraction of its bits. This implies words of the code must be longer than the input strings by adding some redundancy.
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| == Definition ==
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| To measure the distance between two strings, the [[Hamming distance]] is used
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| :<math>\Delta(x,y)=|\{i:x_i \neq y_i\}|</math>
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| The distance of a string <math>w</math> from a code <math>C:\{0,1\}^k\to\{0,1\}^n</math> is computed by
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| :<math>\Delta(w,C)=\min_x\{w,C(x)\}</math>
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| Relative distances are computed as a fraction of the number of bits
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| :<math>\delta(x,y)=\Delta(x,y)/n \text{ and } \delta(w,C)=\Delta(w,C)/n</math>
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| A code <math>C:\{0,1\}^k\to{0,1}^n</math> is called <math>q</math>-local <math>\delta</math>-testable if there exists a Turing machine M given [[random access]] to an input <math>w</math> that makes at most <math>q</math> non-adaptive queries of <math>w</math> and satisfies the following:<ref name=robustLTC>{{cite web|url=http://people.csail.mit.edu/madhu/papers/2004/rltc-conf.pdf|title=Robust Locally Testable Codes and Products of Codes|author=Eli Ben-Sasson, Madhu Sudan}}</ref>
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| :* For any <math>x\in\{0,1\}^k</math> and <math>w=C(x)</math>, <math>Pr[M^w(1^k)=1]=1</math>. In other words, M accepts given access to any codeword of C.
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| :* For <math>w\in\{0,1\}^n</math> such that <math>\delta(w,C)>\delta</math>, <math>Pr[M^w(1^k)=1]\leq1/2</math>. M must reject strings <math>\delta</math>-far from C at least half the time.
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| == Limits ==
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| It remains an open question whether there are any locally testable codes of linear size, but there are several constructions that are considered "nearly linear":<ref name=shortLTC>{{cite web|url=http://eccc.hpi-web.de/eccc-reports/2005/TR05-014/index.html|title=Short Locally Testable Codes and Proofs (Survey)|author=Oded Goldreich}}</ref>
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| # Polynomial arbitrarily close to linear; for any <math>\epsilon>0</math>, <math>n=k^{1+\epsilon}</math>.
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| # Functions of the form <math>n=k^{1+\epsilon(k)}</math>, where <math>\epsilon(k)</math> is a function tending toward 0. This makes n closer to linear as k increases. For example:
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| #* <math>1/\log \log k</math>
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| #* <math>1/(\log k)^c</math> for some <math>c\in (0,1)</math>
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| #* <math>\exp((\log \log \log k)^c)/\log k</math> for <math>c\in (0,1)</math>
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| These have both been achieved, even with constant query complexity and a binary [[Alphabet (computer science)|alphabet]], such as with <math>n=k^{1+1/(\log k)^c}</math> for any <math>c\in (0,1)</math>.
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| The next nearly linear goal is linear up to a [[polylogarithmic]] factor; <math>n=\text{poly}(\log k)*k</math>. Nobody has yet to come up with a linearly testable code that satisfies this constraint.<ref name=shortLTC/>
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| == Parallels ==
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| Locally testable codes have a lot in common with [[probabilistically checkable proofs]] (PCPs). This should be apparent from the similarities of their construction. In both, we are given <math>q</math> random nonadaptive queries into a large string and if we want to accept, we must with probability 1, and if not, we must accept no more than half the time. The major difference is that PCPs are interested in accepting <math>x</math> if there exists a <math>w</math> so that <math>M^w(x)=1</math>. Locally testable codes, on the other hand, accept <math>w</math> if it is part of the code. Many things can go wrong in assuming a PCP proof encodes a locally testable code. For example, the PCP definition says nothing about invalid proofs, only invalid inputs.
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| Despite this difference, locally testable codes and PCPs are similar enough that frequently to construct one, a prover will construct the other along the way.<ref name=LTCandPCP>{{cite web|url=http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.91.9955&rep=rep1&type=pdf|title=Locally Testable Codes|author=Mahdi Cheraghchi-Bashi-Astaneh}}</ref>
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| == Examples ==
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| === Hadamard code ===
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| One of the most famous error-correcting codes, the [[Hadamard code]] is a locally testable code. A codeword x is encoded in the Hadamard code to be the linear function <math>f(y)={\sum_i {x_i y_i}}</math> (mod 2). This requires listing out the result of this function for every possible y, which requires exponentially more bits than its input. To test if a string w is a codeword of the Hadamard code, all we have to do is test if the function it encodes is linear. This means simply checking if <math>w(x)\oplus w(y)=w(x\oplus y)</math> for x and y [[Uniform distribution (discrete)|uniformly random]] vectors (where <math>\oplus</math> denotes [[bitwise XOR]]).
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| It is easy to see that for any valid encoding <math>w</math>, this equation is true, as that is the definition of a linear function. Somewhat harder, however, is showing that a string that is <math>\delta</math>-far from C will have an upper bound on its error in terms of <math>\delta</math>. One bound is found by the direct approach of approximating the chances of exactly one of the three probes yielding an incorrect result. Let A, B, and C be the events of <math>w(x)</math>, <math>w(y)</math>, and <math>w(x\oplus y)</math> being incorrect. Let E be the event of exactly one of these occurring. This comes out to
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| :<math>\begin{align}P(E) &\geq P(A\cup B\cup C)-3*P(A\cup B)\\
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| &\geq 3*P(A)-3*P(A\cup B)-3*P(A\cup B)\\
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| &\geq 3\delta-6\delta^2\end{align}</math>
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| This works for <math>0<\delta\leq5/16</math>, but shortly after, <math>3\delta-6\delta^2<\delta</math>. With additional work, it can be shown that the error is bounded by
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| :<math>f(x) = \begin{cases}
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| 3\delta-6\delta^2 & : 0\leq\delta\leq 5/16 \\
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| 45/128 & : 5/16\leq \delta\leq45/128 \\
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| \delta & :45/128 \leq \delta\leq 1/2
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| \end{cases}
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| </math>
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| For any given <math>\delta</math>, this only has a constant chance of false positives, so we can simply check a constant number of times to get the probability below 1/2.<ref name=Hadamard>{{cite web|url=http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.110.2530&rep=rep1&type=pdf|title=Short Locally Testable Codes and Proofs (Survey)|author=Oded Goldreich}}</ref>
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| === Long code ===
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| The [[Long code]] is another code with very large blowup which is close to locally testable. Given an input <math>0\leq i\leq2^k</math> (note, this takes <math>k</math> bits to represent), the function that returns the <math>i^{th}</math> bit of the input, <math>f_i(x)=x_i</math>, is evaluated on all possible <math>2^k</math>-bit inputs <math>0\leq x\leq2^{2^k}</math>, and the codeword is the concatenation of these (of length <math>n=2^{2^k}</math>). The way to locally test this with some errors is to pick a uniformly random input <math>x</math> and set <math>y=x</math>, but with a small chance of flipping each bit, <math>\mu>0</math>. Accept a function <math>f</math> as a codeword if <math>f(x)=f(y)</math>. If <math>f</math> is a codeword, this will accept <math>f</math> as long as <math>x_i</math> was unchanged, which happens with probability <math>1-\mu</math>. This violates the requirement that codewords are always accepted, but may be good enough for some needs.<ref name=LongCode>{{cite web|url=http://www.wisdom.weizmann.ac.il/~ranraz/publications/Punique_codes.pdf|title=Bounds on Locally Testable Codes with Unique Tests|author=Gillat Kol, Ran Raz}}</ref>
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| Other locally testable codes include [[Reed-Muller codes]] (see [[locally decodable code]]s for a decoding algorithm), [[Reed-Solomon code]]s, and the short code.
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| == References ==
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| {{reflist}}
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| [[Category:Error detection and correction]]
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You just download ready made templates to a separate directory and then choose a favorite one in the admin panel. When you write a new post, you'll see a small bar that goes across the text input area. A Wordpress plugin is a software that you can install into your Wordpress site. They provide many such popular products which you can buy for your baby. Now a days it has since evolved into a fully capable CMS platform which make it, the best platform in the world for performing online business.
The entrepreneurs can easily captivate their readers by using these versatile themes. As of now, Pin2Press is getting ready to hit the market. I've applied numerous Search engine optimization-ready Word - Press themes and I can say from knowledge that I consider the Genesis Search engine marketing panel one particular of the simplest to use. Newer programs allow website owners and internet marketers to automatically and dynamically change words in their content to match the keywords entered by their web visitors in their search queries'a feat that they cannot easily achieve with older software. Premium vs Customised Word - Press Themes - Premium themes are a lot like customised themes but without the customised price and without the wait.
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