en>Paul August: ce, link

2014-01-22T17:13:47Z

ce, link

← Older revision		Revision as of 19:13, 22 January 2014
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	~~Hello~~. ~~Allow me introduce~~ the ~~author~~. ~~Her name~~ is ~~Emilia Shroyer but~~ it's ~~not~~ the ~~most female name out there~~. ~~His family lives~~ in ~~South Dakota~~ but ~~his wife desires~~ them to ~~move~~. ~~Hiring~~ is ~~her day occupation now~~ and ~~she will~~ not ~~alter~~ it ~~whenever quickly~~. ~~To gather cash~~ is ~~one~~ of the ~~things I adore most~~.<br><br>~~Also visit my blog~~ ... [http://~~cn5~~.us/~~healthymealsdelivered68340 meal delivery service~~]		In [[information theory]], an '''erasure code''' is a [[forward error correction]] (FEC) code for the [[binary erasure channel]], which transforms a message of ''k'' symbols into a longer message (code word) with ''n'' symbols such that the original message can be recovered from a subset of the ''n'' symbols. The fraction ''r'' = ''k''/''n'' is called the [[code rate]], the fraction ''k’/k'', where ''k’'' denotes the number of symbols required for recovery, is called [[reception efficiency]].

			==Optimal erasure codes==
			Optimal erasure codes have the property that any ''k'' out of the ''n'' code word symbols are sufficient to recover the original message (i.e., they have optimal reception efficiency). Optimal erasure codes are [[Maximum_distance_separable_code#MDS_codes\|maximum distance separable codes]] (MDS codes).

			Optimal codes are often costly (in terms of memory usage, CPU time, or both) when ''n'' is large. Except for very simple schemes, practical solutions usually have quadratic encoding and decoding [[Computational complexity theory\|complexity]]. Using [[Fast Fourier transform\|FFT]] techniques, the complexity may be reduced to ''O''(''n'' log(''n'')); however, this is not practical.

			===Parity check===
			Parity check is the special case where ''n'' = ''k'' + 1. From a set of ''k'' values <math>\{v_i\}_{1\leq i \leq k}</math>, a check-sum is computed and appended to the ''k'' source values:
			:<math>v_{k+1}= - \sum_{i=1}^k v_i.</math>
			The set of ''k'' + 1 values <math>\{v_i\}_{1\leq i \leq k+1}</math> is now consistent with regard to the check-sum.
			If one of these values, <math>v_e</math>, is erased, it can be easily recovered by summing the remaining variables:
			:<math>v_{e}= - \sum_{i=1 ,i \neq e }^{k+1}v_i.</math>

			===Polynomial oversampling===
			====Example: Err-mail (''k'' = 2)====
			In the simple case where ''k'' = 2, redundancy symbols may be created by sampling different points along the line between the two original symbols. This is pictured with a simple example, called err-mail:

			[[Alice_and_Bob\|Alice]] wants to send her telephone number (555629) to [[Alice_and_Bob\|Bob]] using err-mail. Err-mail works just like e-mail, except
			#About half of all the mail gets lost.<ref>Some versions of this story refer to the err-mail daemon.</ref>
			#Messages longer than 5 characters are illegal.
			#It is very expensive (similar to air-mail).

			Instead of asking Bob to acknowledge the messages she sends, Alice devises the following scheme.

			#She breaks her telephone number up into two parts ''a'' = 555, ''b'' = 629, and sends 2 messages – "''A'' = 555" and "''B'' = 629" – to Bob.
			#She constructs a linear function, <math>f(i) = a + (b-a)(i-1)</math>, in this case <math>f(i) = 555 + 74(i-1)</math>, such that <math>f(1) = 555</math> and <math>f(2) = 629</math>.<br>
			[[Image:Optimal Erasure Codes1.gif]]
			#She computes the values ''f''(3), ''f''(4), and ''f''(5), and then transmits three redundant messages: "C = 703", "D = 777" and "E = 851".

			Bob knows that the form of ''f''(''k'') is <math>f(i) = a + (b-a)(i-1)</math>, where ''a'' and ''b'' are the two parts of the telephone number.
			Now suppose Bob receives "D = 777" and "E = 851".

			[[Image:Optimal Erasure Codes2.gif]]

			Bob can reconstruct Alice's phone number by computing the values of ''a'' and ''b'' from the values (''f''(4) and ''f''(5)) he has received.
			Bob can perform this procedure using any two err-mails, so the erasure code in this example has a rate of 40%.

			Note that Alice cannot encode her telephone number in just one err-mail, because it contains six characters, and the maximum length of one err-mail message is five characters. If she sent her phone number in pieces, asking Bob to acknowledge receipt of each piece, at least four messages would have to be sent anyway (two from Alice, and two acknowledgments from Bob). So the erasure code in this example, which requires five messages, is quite economical.

			<!-- The reader should also note that -->This example is a little bit contrived<!-- in that the linear function itself actually contains Alice's telephone number, i.e., 555 + 74 = 629. This makes the given linear function really only useful for the given data -->. For truly generic erasure codes that work over any data set, we would need something other than the ''f''(''i'') given.

			====General case====
			The linear construction above can be generalized to [[polynomial interpolation]]. Additionally, points are now computed over a [[finite field]].

			First we choose a finite field ''F'' with order of at least ''n'', but usually a power of 2. The sender numbers the data symbols from 0 to ''k'' − 1 and sends them. He then constructs a [[Lagrange polynomial\|(Lagrange) polynomial]] ''p''(''x'') of order ''k'' such that ''p''(''i'') is equal to data symbol ''i''. He then sends ''p''(''k''), ..., ''p''(''n'' − 1). The receiver can now also use polynomial interpolation to recover the lost packets, provided he receives ''k'' symbols successfully. If the order of ''F'' is less than 2<sup>''b''</sup>, where b is the number of bits in a symbol, then multiple polynomials can be used.

			The sender can construct symbols ''k'' to ''n'' − 1 'on the fly', i.e., distribute the workload evenly between transmission of the symbols. If the receiver wants to do his calculations 'on the fly', he can construct a new polynomial ''q'', such that ''q''(''i'') = ''p''(''i'') if symbol ''i'' < ''k'' was received successfully and ''q''(''i'') = 0 when symbol ''i'' < ''k'' was not received. Now let ''r'' = ''p'' − ''q''. Firstly we know that ''r''(''i'') = 0 if symbol ''i'' < ''k'' has been received successfully. Secondly, if symbol ''i'' ≥''k'' has been received successfully, then ''r''(''i'') = ''p''(''i'') − ''q''(''i'') can be calculated. So we have enough data points to construct ''r'' and evaluate it to find the lost packets. So both the sender and the receiver require ''O''(''n'' (''n'' − ''k'')) operations and only ''O''(''n'' − ''k'') space for operating 'on the fly'.

			====Real world implementation====
			This process is implemented by [[Reed–Solomon code]]s, with code words constructed over a [[finite field]] using a [[Vandermonde matrix]].

			==Near-optimal erasure codes==
			''Near-optimal erasure codes'' require (1 + ε)''k'' symbols to recover the message (where ε>0). Reducing ε can be done at the cost of CPU time.
			''Near-optimal erasure codes'' trade correction capabilities for computational complexity: practical algorithms can encode and decode with linear time complexity.

			[[Fountain code]]s (also known as ''rateless erasure codes'') are notable examples of ''near-optimal erasure codes''. They can transform a ''k'' symbol message into a practically infinite encoded form, i.e., they can generate an arbitrary amount of redundancy symbols that can all be used for error correction. Receivers can start decoding after they have received slightly more than ''k'' encoded symbols.

			[[Regenerating code]]s address the issue of rebuilding (also called repairing) lost encoded fragments from existing encoded fragments. This issue arises in distributed
			storage systems where communication to maintain encoded redundancy is a problem.

			==Examples==
			===Near optimal erasure codes===
			*[[Tornado codes]]
			*[[Low-density parity-check codes]]

			===Near optimal fountain (rateless erasure) codes===
			*[[Online codes]]
			*[[LT codes]]
			*[[Raptor codes]]

			===Optimal erasure codes===
			*[[Parity (telecommunication)\|Parity]]: used in [[RAID]] storage systems.
			*[[Parchive]]
			*[[Tahoe-LAFS]] includes [https://pypi.python.org/pypi/zfec zfec]
			*[[Reed–Solomon\|Reed–Solomon coding]]
			* [http://anrg.usc.edu/~maheswaran/Xorbas.pdf Locally Repairable Codes]
			*[http://switzernet.com/people/emin-gabrielyan/051103-erasure-9-5-resilient/ Erasure Resilient Systematic Code], an MDS code outperforming Reed–Solomon in the maximal number of redundant packets, see [http://switzernet.com/people/emin-gabrielyan/051025-erasure-resilient/ RS(4,2) with 2 bits] or [http://switzernet.com/people/emin-gabrielyan/051027-erasure-9-2-resilient/ RS(9,2) with 3 bits]
			*any other [[Maximum_distance_separable_code#MDS_codes\|MDS code]]
			*[[Regenerating Codes]]<ref name="regcodes">{{cite paper \| id = {{citeseerx\|10.1.1.117.6892}} \| title = Original paper }}</ref> see also [http://csi.usc.edu/~dimakis/StorageWiki].

			===Other===
			*[[Spelling alphabet]]

			==See also==
			*[[Forward error correction]] codes.

			==References==
			{{Reflist}}

			==External links==
			*[http://info.iet.unipi.it/~luigi/fec.html Software FEC in computer communications] by Luigi Rizzo describes optimal erasure correction codes
			*[http://feclib.sourceforge.net Feclib] is a near optimal extension to Luigi Rizzo's work that uses band matrices. Many parameters can be set, like the size of the width of the band and size of the finite field. It also successfully exploits the large [[Processor register\|register]] size of modern CPUs. How it compares to the near optimal codes mentioned above is unknown.
			*[http://csi.usc.edu/~dimakis/StorageWiki/doku.php?id=wiki:definitions:repair_problem Coding for Distributed Storage wiki] for regenerating codes and rebuilding erasure codes.

			[[Category:Coding theory]]

en>EmausBot: r2.6.4) (Robot: Adding pl:Funkcja pary

2011-10-23T16:19:35Z

r2.6.4) (Robot: Adding pl:Funkcja pary

New page

Hello. Allow me introduce the author. Her name is Emilia Shroyer but it's not the most female name out there. His family lives in South Dakota but his wife desires them to move. Hiring is her day occupation now and she will not alter it whenever quickly. To gather cash is one of the things I adore most.<br><br>Also visit my blog ... [http://cn5.us/healthymealsdelivered68340 meal delivery service]

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en>Paul August: ce, link

en>EmausBot: r2.6.4) (Robot: Adding pl:Funkcja pary