Determining the number of clusters in a data set: Difference between revisions

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| name          = Long code
| image          =
| image_caption  =
| namesake      =
| type          = [[Block code]]
| block_length  = <math>2^{n}</math> for some <math>n\in\N</math>
| message_length = <math>\log n</math>
| rate          =
| distance      =
| alphabet_size  = <math>2</math>
| notation      = <math>(2^{n},\log n)_2</math>-code
}}
 
In [[theoretical computer science]] and [[coding theory]], the '''long code''' is an [[error detection and correction|error-correcting code]] that is [[locally decodable code|locally decodable]]. Long codes have an extremely poor rate, but play a fundamental role in the theory of [[hardness of approximation]].
 
==Definition==
Let <math>f_1,\dots,f_{2^n} : \{0,1\}^k\to \{0,1\}</math> for <math>k=\log n</math> be the list of ''all'' functions from <math>\{0,1\}^k\to\{0,1\}</math>.
Then the long code encoding of a message <math>x\in\{0,1\}^k</math> is the string <math>f_1(x)\circ f_2(x)\circ\dots\circ f_{2^n}(x)</math> where <math>\circ</math> denotes concatenation of strings.
This string has length <math>2^n=2^{2^k}</math>.
 
The [[Walsh-Hadamard code]] is a subcode of the long code, and can be obtained by only using functions <math>f_i</math> that are [[linear function]]s when interpreted as functions <math>\mathbb F_2^k\to\mathbb F_2</math> on the [[finite field]] with two elements. Since there are only <math>2^k</math> such functions, the block length of the Walsh-Hadamard code is <math>2^k</math>.
 
An equivalent definition of the long code is as follows:
The Long code encoding of <math>j\in[n]</math> is defined to be the truth table of the Boolean dictatorship function on the <math>j</math>th coordinate, i.e., the truth table of <math>f:\{0,1\}^n\to\{0,1\}</math> with <math>f(x_1,\dots,x_n)=x_i</math>.<ref>Definition 7.3.1 in [http://arxiv.org/abs/1002.3864  Limits of Approximation Algorithms: PCPs and Unique Games (DIMACS Tutorial Lecture Notes)]</ref>
Thus, the Long code encodes a <math>(\log n)</math>-bit string as a <math>2^n</math>-bit string.
 
==Properties==
The long code does not contain repetitions, in the sense that the function <math>f_i</math> computing the <math>i</math>th bit of the output is different from any function <math>f_j</math> computing the <math>j</math>th bit of the output for <math>j\neq i</math>.
Among all codes that do not contain repetitions, the long code has the longest possible output.
Moreover, it contains all non-repeating codes as a subcode.
 
==References==
{{reflist}}
 
[[Category:Coding theory]]
[[Category:Error detection and correction]]

Latest revision as of 07:36, 15 May 2014

Who is your competitors? Could it be the box shop down the street that will promote similar products...

Who is my competition, if you are on web business manager you should consider? If youre like every other self-respecting company owner you"ve searched the Internet finding other sites that sell similar merchandise or services. And like the majority of companies you learn what you can from these web sites and make use of what you learn with regards to items, pricing and demonstration.

Who"s your competitors? Is it the box store down similar products that can be sold by the street for under you can purchase yours? If youre similar to business people you find creative methods to market your product in a way that highlights the initial features and characteristics that cant be duplicated in old-fashioned box shop purchases.

But are box shops your rivals? Are other online retailers your competitors? Who is your competitors?

You"re! You"re your own competition.

As a business owner you will need to pay attention to what other organizations do (both on line and in stone and mortar counterparts). In the long run, nevertheless, you"re in competition with yourself first and foremost. Learn more on our favorite partner paper by clicking fundable ledified.

You may or may perhaps not manage to match or beat yet another companies in cost, but you can work hard to do better running a business this year than you did last by supplying more competitive pricing or excellent customer-service. Work hard to outperform your-self.

Think for a moment about all the things that you will be doing better at this year than last. Visit ledified fundable to learn the inner workings of this belief. Dont limit your thinking to financial information.

Are you better able to handle administrative duties this season than last?

Have you been creating new networks between customers and business associates?

Are you implementing new marketing strategies that you didnt find out about last year?

Have you been streamlining customer support by developing autoresponders and faster product pleasure choices?

Are you making sure your website remains fresh and appealing in a way that you werent able to last year?

Are you more excited about your organization this year as you were last year?

Are you eliminating others that not offer as well and answering client needs for particular items?

Are you standing still or are you using ways of forward progress?

I might ask many additional questions, but the point of this exercise would be to encourage you to pay more hours fighting with your-self than with another business. If you get this to your goal you will usually discover greater business success.

You"re better able to understand what you can from problems youve manufactured in the past and increase for the future when you compete against yourself. You"ve (or are) making when you focus too much on what someone else is performing you"re less in a position to evaluate what mistakes. In the end you may well not have the ability to be objective about how precisely you"re doing as a business because you"re paying an excessive amount of time concentrating on a business you don"t own. Discover further on our favorite related URL by browsing to UNISDR - Regional office for the Americas.

Perhaps here is the year you can begin a fresh and healthy competition with your self..

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