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In [[coding theory]], the '''weight enumerator polynomial''' of a binary [[linear code]] specifies the number of words of each possible  [[Hamming weight]].
== I want to ask you a few very embarrassing thing ==


Let <math>C \subset \mathbb{F}_2^n</math> be a binary linear code length <math>n</math>. The '''weight distribution''' is the sequence of numbers
'Leave him ...... I'll ask you one, you really want the crooks to justice it?' Li Mei [http://www.dmwai.com/webalizer/kate-spade-11.html ケイトスペード 財布 通販] asked.<br><br>'Of course ...... I can not wait to kill him, I told him so well, are [http://www.dmwai.com/webalizer/kate-spade-8.html マザーズバッグ ケイトスペード] willing to put heart to him, but he was to tens of thousands of dollars, lie to [http://www.dmwai.com/webalizer/kate-spade-13.html ケイトスペード 人気バッグ] me ...... woo.' Yang Ye grief, all of a sudden wail scrambled.<br><br>Li Mei stroking her shoulders, straight was crying a little lighter, a small channel with her: 'This play with emotion, he should be duly punished, we have found clues ...... Next, I want to ask you a few very embarrassing thing, you are willing to talk to me? very likely through this clue to find him. '<br><br>Yang Ye wiping away tears, looked slightly puzzled Li Mei, [http://www.dmwai.com/webalizer/kate-spade-5.html ケイトスペード バッグ アウトレット] also look jing police stationed outside the door for so much as her busy, she felt faint heart touched, wipe tears, calm way forward : 'Sister, I believe you ...... you ask it, to the last, and I have nothing embarrassing.'<br><br>'your trust, there will be [http://www.dmwai.com/webalizer/kate-spade-0.html ハンドバッグ ケイトスペード] a return, I want to ask is that you xing thing .........'
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:<math> A_t = \#\{c \in C \mid w(c) = t \} </math>
== For example ==


giving the number of [[codeword]]s ''c'' in ''C'' having weight ''t'' as ''t'' ranges from 0 to ''n''.  The '''weight enumerator''' is the bivariate [[polynomial]]
Like ...... just [http://www.dmwai.com/webalizer/kate-spade-10.html ケイトスペード 財布 セール] like the first hand the kind of bully, one night, stealing more than a [http://www.dmwai.com/webalizer/kate-spade-5.html ケイトスペード バッグ 激安] dozen plates and give you a call on the window, to retrieve the license plate is not it, for a hundred dollars and I tell you where to find, than you save more ...... go through a driver for fear of difficulty mentality, a lot of really fighting. [http://www.dmwai.com/webalizer/kate-spade-13.html ケイトスペードショルダーバッグ新作] Encounter disloyal to you, met disloyal, no amount to fifty, you say give or not to give? Chain fraud, Mozhun victim mentality. Xiaomeng [http://www.dmwai.com/webalizer/kate-spade-15.html ケイトスペード マザーズバッグ] Qi wondering, this case seems more like an upgraded version of Xian Rentiao, and added to the anesthesia robbery, fraud added to the secondary, compared to the old-fashioned hand, [http://www.dmwai.com/webalizer/kate-spade-15.html ケイトスペード クラッチバッグ] more technology, more hidden spacious.<br><br>For example, on one side so hurry, and after careful makeup, but even the victim can not tell the suspect's facial features accurately.<br><br>do not, big bowls dumbfounded at this time, emerged on the computer screen 骆家龙 body as seven or eight women, all of a sudden see this cargo bite the hand index finger, mouth Flows, a pair of gentlemen look.<br><br>'carefully identify what, there are no suspects you've ever seen.' Luo Jialong politely asked.
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</ul>


:<math> W(C;x,y) = \sum_{w=0}^n A_w x^w y^{n-w}.</math>
== so I should be directly put ==


==Basic properties==
Surprisingly, riding a broken bike, it was said that he show, but had denied doing so many shows are really not easy, many students long after graduation, back to school looking at [http://www.dmwai.com/webalizer/kate-spade-13.html kate spade ハンドバッグ] old principal remains the same, but also Christine respectful King to call a 'Wang.'<br><br>Yes, he was once a professional criminal investigation traces test instructor, Zhibian two decades, under the [http://www.dmwai.com/webalizer/kate-spade-8.html マザーズバッグ ケイトスペード] command of the police in a lot of celebrities out, such as a detective in the name of Xu Pingqiu, such as [http://www.dmwai.com/webalizer/kate-spade-0.html ケイトスペード中古バッグ] the memory of the heroes Shao [http://www.dmwai.com/webalizer/kate-spade-9.html ケイトスペードバッグセール] Bing province's mountains, such as are still Council Secretary bits are his students.<br><br>he stood up, [http://www.dmwai.com/webalizer/kate-spade-8.html マザーズバッグ ケイトスペード] accustomed to standing speech, who rarely led shelf, he took the microphone, looked a current student, always unscripted speech, has been talking about his life, and he cleared his throat and said:<br><br>'Students, remember the entrance ceremony, my first words yes, I am your president Wang Lan, congratulations admitted to the police academy.<br><br>Today, I am still president, you will no longer be students, so I should be directly put, Congratulations, graduate. Although many people are under false pretenses, and
#<math> W(C;0,1) = A_{0}=1 </math>
相关的主题文章:
#<math> W(C;1,1) = \sum_{w=0}^{n}A_{w}=|C| </math>
<ul>
#<math> W(C;1,0) = A_{n}= 1 \mbox{ iff } (1,\ldots,1)\in C\ \mbox{ and } 0 \mbox{ otherwise.} </math>
 
#<math> W(C;1,-1) = \sum_{w=0}^{n}A_{w}(-1)^{n-w} = A_{n}+(-1)^{1}A_{n-1}+\ldots+(-1)^{n-1}A_{1}+(-1)^{n}A_{0} </math>
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</ul>


==MacWilliams identity==
== helpless authentic sentence ==
Denote the [[dual code]] of <math>C \subset \mathbb{F}_2^n</math> by


:<math>C^\perp = \{x \in \mathbb{F}_2^n \,\mid\, \langle x,c\rangle = 0 \mbox{ }\forall c \in C \} </math>  
Not convinced, give Shao Okavango call, but the phone said a few words, dejected expression on his face is very strong, Dong Shaojun know to die, he answered the phone, whispered a few words, and then snapped a press phone jack throw to Wu Guangyu, helpless authentic sentence:<br><br>'slut, you win.'<br><br>'Hey Hey, let's go.' I sin of pride authentic.<br><br>Wu Guangyu slightly reluctantly, and return the car to the fourth prison.<br><br>homework that day, but spared from the afternoon talking about the evening, and then [http://www.dmwai.com/webalizer/kate-spade-8.html kate spade 財布 ゴールド] there took place in the Arabian Nights, and even talk about discipline in the department is also impatient, until late at night a few people left fourth prison, but from the thief mouth where to get more names and nicknames, cattle traders, selling [http://www.dmwai.com/webalizer/kate-spade-3.html ケイトスペード 財布 新作] veterinary drugs, mule animal fur market and the closing of this strange world, in gradually to [http://www.dmwai.com/webalizer/kate-spade-13.html ケイトスペード 人気バッグ] several small police did not know to start his hideous face when they once again set foot [http://www.dmwai.com/webalizer/kate-spade-15.html ケイトスペード マザーズバッグ] on search time, not become strong and courageous, but rather become more hesitant and confused.<br><br>next [http://www.dmwai.com/webalizer/kate-spade-3.html ケイトスペード リボン バッグ] morning, another accident occurred, and a
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(where <math><,></math> denotes the vector [[dot product]] and which is taken over <math>\mathbb{F}_2</math>).
== dumbfounding to cater to the boss ==


The '''MacWilliams identity''' states that
The. 'Xu Pingqiu laughed.<br><br>more and more outrageous, Shike Zhang simply do not ask, but look Xu Director so got the idea, he was a little laughing, such as a cigarette effort, they appear to pick fights student's car out, is a large body of work Security Section, the two maintained did not move, watching the cars go out very far, just to twist the key Shike Zhang launch, only to be promised at number one pressed, he looked gestured: 'Look, say what was [http://www.dmwai.com/webalizer/kate-spade-9.html ケイトスペード リボン バッグ] . '<br><br>Hey Hey, gate driving fast in a few bikes out, chasing [http://www.dmwai.com/webalizer/kate-spade-3.html ケイトスペード バッグ 通販] the car in the direction to walk in sin is more than most before picking [http://www.dmwai.com/webalizer/kate-spade-12.html ケイトスペード ハンドバッグ] up the ass of a pull people, a dozen, it looks like not seek trouble-shirts are there will be something else.<br><br>'Let the periphery, holiday tune several of today's field, to see what they were doing. suck, do not look at me, this is for security reasons, in case these Lengxiao Zi did not re-play of light to how to [http://www.dmwai.com/webalizer/kate-spade-13.html ケイトスペード バッグ ショルダー] do things? '<br><br>Xu Pingqiu way forward, pretending to be serious, but the taste is very strong to see a joke, Shike Zhang be silent, dumbfounding to cater to [http://www.dmwai.com/webalizer/kate-spade-15.html ケイトスペード マザーズバッグ] the boss, touched the phone, United
 
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:<math>W(C^\perp;x,y) = \frac{1}{\mid C \mid} W(C;y-x,y+x). </math>
<ul>
 
 
The identity is named after [[Jessie MacWilliams]].
  <li>[http://61.189.52.138/plus/view.php?aid=521308 http://61.189.52.138/plus/view.php?aid=521308]</li>
 
 
==Distance enumerator==
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The '''distance distribution''' or '''inner distribution''' of a code ''C'' of size ''M'' and length ''n'' is the sequence of numbers
 
 
  <li>[http://www.wowodao.com/forum.php?mod=viewthread&tid=349934 http://www.wowodao.com/forum.php?mod=viewthread&tid=349934]</li>
:<math> A_i = \frac{1}{M} \# \left\lbrace (c_1,c_2) \in C \times C \mid d(c_1,c_2) = i \right\rbrace </math>
 
 
</ul>
where ''i'' ranges from 0 to ''n''.  The '''distance enumerator polynomial''' is  
 
:<math> A(C;x,y) = \sum_{i=0}^n A_i x^i y^{n-i} </math>
 
and when ''C'' is linear this is equal to the weight enumerator.
 
The '''outer distribution''' of ''C'' is the 2<sup>''n''</sup>-by-''n''+1 matrix ''B'' with rows indexed by elements of GF(2)<sup>''n''</sup> and columns indexed by integers 0...''n'', and entries
 
:<math> B_{x,i} = \# \left\lbrace c \in C \mid d(c,x) = i \right\rbrace . </math>
 
The sum of the rows of ''B'' is ''M'' times the inner distribution vector (''A''<sub>0</sub>,...,''A''<sub>''n''</sub>).
 
A code ''C'' is '''regular''' if the rows of ''B'' corresponding to the codewords of ''C'' are all equal.
 
==References==
* {{cite book | last=Hill | first=Raymond | title=A first course in coding theory | publisher=[[Oxford University Press]] | series=Oxford Applied Mathematics and Computing Science Series | date=1986 | isbn=0-19-853803-0 | pages=165–173 }}
* {{cite book | last = Pless | first = Vera | authorlink=Vera Pless | title = Introduction to the theory of error-correcting codes | publisher = [[John Wiley & Sons]]|series = Wiley-Interscience Series in Discrete Mathematics | date = 1982| isbn = 0-471-08684-3 | pages=103–119 }}
* {{cite book | author=J.H. van Lint | title=Introduction to Coding Theory | edition=2nd ed | publisher=[[Springer-Verlag]] | series=[[Graduate Texts in Mathematics|GTM]] | volume=86 | date=1992 | isbn=3-540-54894-7 }} Chapters 3.5 and 4.3.
 
[[Category:Coding theory]]
[[Category:Error detection and correction]]
[[Category:Mathematical identities]]

Latest revision as of 08:20, 19 August 2014

I want to ask you a few very embarrassing thing

'Leave him ...... I'll ask you one, you really want the crooks to justice it?' Li Mei ケイトスペード 財布 通販 asked.

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Yang Ye wiping away tears, looked slightly puzzled Li Mei, ケイトスペード バッグ アウトレット also look jing police stationed outside the door for so much as her busy, she felt faint heart touched, wipe tears, calm way forward : 'Sister, I believe you ...... you ask it, to the last, and I have nothing embarrassing.'

'your trust, there will be ハンドバッグ ケイトスペード a return, I want to ask is that you xing thing .........' 相关的主题文章:

For example

Like ...... just ケイトスペード 財布 セール like the first hand the kind of bully, one night, stealing more than a ケイトスペード バッグ 激安 dozen plates and give you a call on the window, to retrieve the license plate is not it, for a hundred dollars and I tell you where to find, than you save more ...... go through a driver for fear of difficulty mentality, a lot of really fighting. ケイトスペードショルダーバッグ新作 Encounter disloyal to you, met disloyal, no amount to fifty, you say give or not to give? Chain fraud, Mozhun victim mentality. Xiaomeng ケイトスペード マザーズバッグ Qi wondering, this case seems more like an upgraded version of Xian Rentiao, and added to the anesthesia robbery, fraud added to the secondary, compared to the old-fashioned hand, ケイトスペード クラッチバッグ more technology, more hidden spacious.

For example, on one side so hurry, and after careful makeup, but even the victim can not tell the suspect's facial features accurately.

do not, big bowls dumbfounded at this time, emerged on the computer screen 骆家龙 body as seven or eight women, all of a sudden see this cargo bite the hand index finger, mouth Flows, a pair of gentlemen look.

'carefully identify what, there are no suspects you've ever seen.' Luo Jialong politely asked. 相关的主题文章:

so I should be directly put

Surprisingly, riding a broken bike, it was said that he show, but had denied doing so many shows are really not easy, many students long after graduation, back to school looking at kate spade ハンドバッグ old principal remains the same, but also Christine respectful King to call a 'Wang.'

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'Students, remember the entrance ceremony, my first words yes, I am your president Wang Lan, congratulations admitted to the police academy.

Today, I am still president, you will no longer be students, so I should be directly put, Congratulations, graduate. Although many people are under false pretenses, and 相关的主题文章:

helpless authentic sentence

Not convinced, give Shao Okavango call, but the phone said a few words, dejected expression on his face is very strong, Dong Shaojun know to die, he answered the phone, whispered a few words, and then snapped a press phone jack throw to Wu Guangyu, helpless authentic sentence:

'slut, you win.'

'Hey Hey, let's go.' I sin of pride authentic.

Wu Guangyu slightly reluctantly, and return the car to the fourth prison.

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dumbfounding to cater to the boss

The. 'Xu Pingqiu laughed.

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