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| {{infobox code
| | For the people at all like me which are not very techy, the look at the standards therefore announces about the earth-friendly component enter is distributed as being an AV online video tips. Connect the particular outdated audio-video pink (online video) connection in and plug this yellow and also white audio connections for the audio stimulant. And so I determined i possibly could go with a letter lead per regarding the audio stimulant through the specific VCR and also DVR ( accustomed use digital result) along with a letter cable [http://www.youtube.com/watch?v=fw6v5RGfkCQ lcd tv reviews] during the alternative component input which will experience the eco friendly part and also yellow Ab comment. I was thinking for as long as e failed to obtain each of gadgets activated in addition the specific "joint " design must move. Fine, the most important audio is compatible however since the Television "senses" gold watches what exactly is rocked in, it will use the specific DVR (ingredient) yet not in the event that VCR (blue online) is just fired up. This tool "feels" there are pair alternative part cabling blocked as part of and so [http://www.youtube.com/watch?v=3OPlx_X2hZc 32 lcd] provides and also "invalid style" since it plans a different element cable tv. This means you really have to plug as well as unplug each cables dependent on that you wish to use - isn't planning arise. Some of the DVR wore audio-video outputs not to mention part and so required incorporate they in the specific letter lead install understanding that is compatible. And so I've forgotten the product quality of the point outputs from the DVR. <br><br> |
| | name = Binary Justesen Codes
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| | image =
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| | image_caption =
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| | namesake = Jørn Justesen
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| | type = [[Linear block code]]
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| | block_length = <math>n</math>
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| | message_length = <math>k</math>
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| | rate = <math>R=k/n</math>
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| | distance = <math>\delta n</math> where <math>\delta\geq \Big(1-R-\epsilon\Big) H^{-1}_2\big(\frac{1}{2}-\epsilon\big) \sim \frac{1}{2}(1-R-\epsilon)</math> for small <math>\epsilon>0</math>.
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| | alphabet_size = 2
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| | notation = <math>\left[ n, k, \delta n \right]_2</math>-code
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| | properties = constant rate, constant relative distance, constant alphabet size
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| }}
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| In [[coding theory]], '''Justesen codes''' form a class of [[Error detection and correction|error-correcting codes]] that have a constant rate, constant relative distance, and a constant alphabet size.
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| Before the Justesen code was discovered, no code was known that had all of these three parameters as a constant.
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| Subsequently, other codes with this property have been discovered, for example [[expander code]]s.
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| These codes have important applications in [[computer science]] such as in the construction of [[small-bias sample space]]s.
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| Justesen codes are derived as the [[Concatenated error correction code|code concatenation]] of a [[Reed–Solomon error correction|Reed–Solomon code]] and the [[Wozencraft ensemble]].
| | Downsides: Extremely, few. My personal nothing but essential problem would be that whilst i the increase bandstand, these products aren't the fact that supporting. When I hit some of the display in the top, it'll careen backward and forward a little bit of as a result of thighs certainly not granting proper influence. When you have a well balanced stand comparable to I choose, this is certainly virtually the none distribute, however it may well be a worries towards some. Absolutely everyone feedback as to how that they like the style for its cut still (personally added). Additionally, i had really like any more HIGH-DEFINITION MULTIMEDIA INTERFACE position, then again two is perfect for a lot of makes use of. In addition, i am I am fairly technical savy, however i'd obtain cherished your user guide which goes into better discourse as well as level. Consistent regarding TV which is very bare osseins, there are numerous choices to adjust and also the manual hardly addresses whatsoever. Your lengthier guide may have been extremely ready. You will have to chub mackerel on it one more. Certainly not a main question. <br><br> Now. I've had this for around 3 several weeks then. the has gone from using a Insignia 37" 60hz no-clever Liquid Crystal Display to the Tv programs. Its in our den, so our very own principal Tv programs. e was happier with our personal Insignia, but certainly essential something heavier when I might get headaches during the night, with perspective striving. My own mom has experienced the Vizio for many age it has supported the potentially. And once e watched i could possibly use 12 missouri desire f-r-e-e the following regarding The amazon online marketplace, i assumed i'd begin our browse. This time the calculate every little thing around enough related with paralysis by simply over basic research, by checking out lead studies (on the amazon website and AVSforum) and additionally support studies with the wants concerning CNET. Unfortunately this Television is simply new CNET needs an appraisal off but (with luck , early). Recommended.net has the scientific assessment from the specific 48" unit tho. |
| The Reed–Solomon codes used achieve constant rate and constant relative distance at the expense of an alphabet size that is ''linear'' in the message length.
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| The [[Wozencraft ensemble]] is a family of codes that achieve constant rate and constant alphabet size, but the relative distance is only constant for most of the codes in the family.
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| The concatenation of the two codes first encodes the message using the Reed–Solomon code, and then encodes each symbol of the codeword further using a code from the [[Wozencraft ensemble]] – using a different code of the ensemble at each position of the codeword.
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| This is different from usual code concatenation where the inner codes are the same for each position.
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| The Justesen code can be can constructed very efficiently using only [[logarithmic space]].
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| ==Definition==
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| Justesen code is [[Concatenated error correction code|concatenation code]] with different linear inner codes, which is composed of an <math>(N,K,D)_{q^k}</math> outer code <math>C_{out}</math> and different <math>(n,k,d)_q</math> inner codes <math>C_{in}^i</math>, <math>1 \le i \le N</math>. More precisely, the concatenation of these codes, denoted by <math>C_{out} \circ (C_{in}^1 ,...,C_{in}^N )</math>, is defined as follows. Given a message <math>m \in [q^k]^K</math>, we compute the codeword produced by an outer code <math>C_{out}</math>: <math>C_{out}(m) = (c_1,c_2,..,c_N)</math>. Then we apply each code of N linear inner codes to each coordinate of that codeword to produce the final codeword; that is, <math>C_{out} \circ (C_{in}^1,..,C_{in}^N)(m) = (C_{in}^1(c_1),C_{in}^2(c_2),..,C_{in}^n(c_N))</math>. Look back to the definition of the outer code and linear inner codes, this definition of the Justesen code makes sense because the codeword of the outer code is a vector with <math>N</math> elements, and we have <math>N</math> linear inner codes to apply for those <math>N</math> elements.
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| Here for the Justesen code, the outer code <math>C_{out}</math> is chosen to be [[Reed–Solomon error correction|Reed Solomon code]] over a [[Field (mathematics)|field]] <math>\mathbb{F}_{q^k}</math> evaluated over <math>\mathbb{F}_{q^k}-\{ 0 \}</math> of [[Code rate|rate]] <math>R</math>, <math>0</math> < <math>R</math> < <math>1</math>. The outer code <math>C_{out}</math> have the relative distance <math>\delta_{out} = 1 - R</math> and block length of <math>N = q^k-1</math>. The set of inner codes is the [[Wozencraft ensemble]] <math>\{ C_{in}^\alpha \} _{\alpha \in \mathbb{F}_{q^k }^* }</math>.
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| ==Property of Justesen Code==
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| As the linear codes in the Wonzencraft ensemble have the rate <math>\frac{1}{2}</math>, Justesen code is the concatenated code <math>C^* = C_{out} \circ (C_{in}^1,C_{in}^2,..,C_{in}^N)</math> with the rate <math>\frac{R}{2}</math>. We have the following theorem that estimates the distance of the concatenated code <math>C^*</math>.
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| ==Theorem==
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| Let <math>\varepsilon</math> > 0. <math>C^*</math> has relative distance at least <math>(1-R-\varepsilon)H_q^{-1}(\frac{1}{2}-\varepsilon)</math>.
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| '''''Proof:'''''
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| The idea of proving that the code <math>C^*</math> has the distance at least <math>(1-R-\varepsilon)H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k \cdot N</math> is to prove that the Hamming distance of two different codewords is at least <math>(1-R-\varepsilon)H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k \cdot N</math>.
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| Denote <math>\Delta(c^1,c^2)</math> be the Hamming distance of two codewords <math>c^1</math> and <math>c^2</math>.
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| So for any given <math>m_1</math> and <math>m_2</math> in <math>(\mathbb{F}_{q^k})^K</math> (<math>m_1 \ne m_2</math>), we want to lower bound <math>\Delta(C^*(m_1),C^*(m_2))</math>.
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| Notice that if <math>C_{out}(m) = (c_1,c_2,..,c_N)</math>, then <math>C^*(m) = (C_{in}^1(c_1),C_{in}^2(c_2),..,C_{in}^N(c_N))</math>. So to the lower bound <math>\Delta(C^*(m_1),C^*(m_2))</math>, we need to take into account the distance of <math>C_{in}^i</math> for i = 1,2,…,N.
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| Suppose <math>C_{out}(m_1) = (c_1^1,c_2^1,..,c_N^1)</math> and <math>C_{out}(m_2) = (c_1^2,c_2^2,..,c_N^2)</math>.
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| Recall that <math>\{ C_{in}^i \}_{1 \le i \le N}</math> is a [[Wozencraft ensemble]]. Due to "Wonzencraft ensemble theorem", there are at least <math>(1-\varepsilon)N</math> linear codes <math>C_{in}^i</math> that have distance <math>H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k</math>.
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| So if for some <math>1 \le i \le N</math>, <math>c_i^1 \ne c_i^2</math> and <math>C_{in}^i</math> code has distance <math>\ge H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k</math>, then <math>\Delta(C_{in}^i(c_i^1),C_{in}^i(c_i^2)) \ge H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k</math>.
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| Further, if we have <math>T</math> numbers <math>1 \le i \le N</math> such that <math>c_i^1 \ne c_i^2</math> and <math>C_{in}^i</math> code has distance <math>\ge H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k</math>, then <math>\Delta(C^*(m_1),C^*(m_2)) \ge H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k \cdot T</math>.
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| So now the final task is to lower bound <math>T</math>.
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| Denote S be the set of all <math>i</math> (<math>1 \le i \le N</math>) such that <math>c_i^1 \ne c_i^2</math>. Then <math>T</math> is the number of linear codes <math>C_{in}^i</math> (<math>i \in S</math>) having the distance <math>H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k</math>.
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| Now we want to estimate <math>\left| S \right|</math>. Obviously <math>\left| S \right| = \Delta(C_{out}(m_1),C_{out}(m_2)) \ge (1-R)N</math>. | |
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| Due to the [[Wozencraft ensemble|Wozencraft Ensemble Theorem]], there are at most <math>\varepsilon N</math> linear codes having distance less than <math>H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k</math>, so <math>T \ge \left| S \right| - \varepsilon N \ge (1-R)N - \varepsilon N = (1-R-\varepsilon)N</math>.
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| Finally,we have
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| <math>\Delta(C^*(m_1),C^*(m_2)) \ge H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k \cdot T \ge H_q^{-1}(\frac{1}{2}-\varepsilon) \cdot 2k \cdot (1-R-\varepsilon) \cdot N</math>.
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| This is true for any arbitrary <math>m_1 \ne m_2</math>. So <math>C^*</math> has the relative distance at least <math>(1-R-\varepsilon)H_q^{-1}(\frac{1}{2}-\varepsilon)</math>, which completes the proof.
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| ==Comments==
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| We want to consider the "strongly explicit code". So the question is what the "strongly explicit code" is. Loosely speaking, for linear code, the "explicit" property is related to the complexity of constructing its generator matrix G. That means, we can compute the matrix in logarithmic space without using the brute force algorithm to verify that a code has a given satisfied distance.
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| For the other codes that are not linear, we can consider the complexity of the encoding algorithm.
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| So by far, we can see that the Wonzencraft ensemble and Reed-Solomon codes are strongly explicit. Therefore we have the following result:
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| '''''Corollary:''''' The concatenated code <math>C^*</math> is an asymptotically good code(that is, rate <math>R</math> > 0 and relative distance <math>\delta</math> > 0 for small q) and has a strongly explicit construction.
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| ==An example of a Justesen code==
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| The following slightly different code is referred to as the Justesen code in MacWilliams/MacWilliams. It is the particular case of the above-considered
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| Justesen code for a very particular Wonzencraft ensemble:
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| Let ''R'' be a Reed-Solomon code of length ''N'' = 2<sup>''m''</sup> − 1, [[dimension (vector space)|rank]] ''K'' and minimum weight ''N'' − ''K'' + 1. The symbols of ''R'' are elements of ''F'' = GF(2<sup>''m''</sup>) and the codewords are obtained by taking every polynomial ƒ over ''F'' of degree less than ''K'' and listing the values of ƒ on the non-zero elements of ''F'' in some predetermined order. Let α be a [[primitive element (finite field)|primitive element]] of ''F''. For a codeword '''a''' = (''a''<sub>1</sub>, ..., ''a''<sub>''N''</sub>) from ''R'', let '''b''' be the vector of length 2''N'' over ''F'' given by
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| :<math> \mathbf{b} = \left( a_1, a_1, a_2, \alpha^1 a_2, \ldots, a_N, \alpha^{N-1} a_N \right) </math>
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| and let '''c''' be the vector of length 2''N'' ''m'' obtained from ''b'' by expressing each element of ''F'' as a binary vector of length ''m''. The ''Justesen code'' is the linear code containing all such '''c'''. | |
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| The parameters of this code are length 2''m'' ''N'', dimension ''m'' ''K'' and [[minimum distance]] at least
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| :<math> \sum_{i=1}^\ell i \binom{2m}{i} , </math>
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| where <math>\ell</math> is the greatest integer satisfying <math>\sum_{i=1}^\ell \binom{2m}{i}\leq N-K+1</math>. (See MacWilliams/MacWilliams for a proof.)
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| ==See also==
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| # [[Wozencraft ensemble]]
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| # [[Concatenated error correction code]]
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| # [[Reed–Solomon error correction|Reed-Solomon error correction]]
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| # [[Linear code|Linear Code]]
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| ==References==
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| # [http://www.cse.buffalo.edu/~atri/courses/coding-theory/ Lecture 28: Justesen Code. Coding theory's course. Prof. Atri Rudra].
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| # [http://people.csail.mit.edu/madhu/FT02/ Lecture 6: Concatenated codes. Forney codes. Justesen codes. Essential Coding Theory].
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| # {{cite journal | author=J. Justesen | title=A class of constructive asymptotically good algebraic codes | journal=IEEE Trans. Info. Theory | volume=18 | year=1972 | pages=652–656 | doi=10.1109/TIT.1972.1054893 | issue=5 }}
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| # {{cite book | author=F.J. MacWilliams | authorlink=Jessie MacWilliams | coauthors=N.J.A. Sloane | title=The Theory of Error-Correcting Codes | publisher=North-Holland | year=1977 | isbn=0-444-85193-3 | pages=306–316 }}
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| [[Category:Error detection and correction]]
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| [[Category:Finite fields]]
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| [[Category:Coding theory]]
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For the people at all like me which are not very techy, the look at the standards therefore announces about the earth-friendly component enter is distributed as being an AV online video tips. Connect the particular outdated audio-video pink (online video) connection in and plug this yellow and also white audio connections for the audio stimulant. And so I determined i possibly could go with a letter lead per regarding the audio stimulant through the specific VCR and also DVR ( accustomed use digital result) along with a letter cable lcd tv reviews during the alternative component input which will experience the eco friendly part and also yellow Ab comment. I was thinking for as long as e failed to obtain each of gadgets activated in addition the specific "joint " design must move. Fine, the most important audio is compatible however since the Television "senses" gold watches what exactly is rocked in, it will use the specific DVR (ingredient) yet not in the event that VCR (blue online) is just fired up. This tool "feels" there are pair alternative part cabling blocked as part of and so 32 lcd provides and also "invalid style" since it plans a different element cable tv. This means you really have to plug as well as unplug each cables dependent on that you wish to use - isn't planning arise. Some of the DVR wore audio-video outputs not to mention part and so required incorporate they in the specific letter lead install understanding that is compatible. And so I've forgotten the product quality of the point outputs from the DVR.
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