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| {{Inadequate lead|date=March 2013}}
| | == there are lotus. == |
| {{Use dmy dates|date=September 2011}}
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| The theory of '''association schemes''' arose in statistics, in the theory of [[design of experiments|experimental design]] for the [[analysis of variance]].<ref>{{harvnb|Bailey|2004|loc=pg. 387}}</ref><ref>{{harvnb|Bose|Mesner|1959}}</ref><ref>{{harvnb|Bose|Nair|1939}}</ref> In [[mathematics]], association schemes belong to both [[algebra]] and [[combinatorics]]. Indeed, in [[algebraic combinatorics]], association schemes provide a unified approach to many topics, for example [[combinatorial design]]s and [[coding theory]].<ref>{{harvnb|Bannai|Ito|1984}}</ref><ref>{{harvnb|Godsil|1993}}</ref> In algebra, association schemes generalize [[group (mathematics)|group]]s, and the theory of association schemes generalizes the [[group character|character theory]] of [[group representation|linear representations]] of groups.<ref>{{harvnb|Bailey|2004|loc=pg. 387}}</ref><ref>{{harvnb|Zieschang|2005b}}</ref><ref>{{harvnb|Zieschang|2005a}}</ref>
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| ==Definition== | | In a lot of colorful lotus land records.<br><br>ordinary barrel. Also half-human Heights barrel. Half a bucket can be.<br><br>and this pond. Nearly circular shape. About nearly two meters in diameter, so big pond. Inside the one yuan heavy water. Estimated to be sufficient to store ten barrels of heavy water to one yuan.<br><br>both ends of [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_1.php クリスチャンルブタン 靴 メンズ] the pond in the [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン バッグ] yuan heavy water. Zhang Lin just two colorful, colorful lotus really look the same as [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_4.php クリスチャンルブタン セール] described in the reel. Lotus is divided into 288, lotus minimal, there are lotus.<br><br>'Huh?' Qin [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン 靴] Yu suddenly surprised again.<br><br>'central pond is arranged a little to hide the magic circle.' Qin Yu force to explore space discovered. This pond central in the end what is being used to hide the magic array block.<br><br>the passage of time, then quickly passed more than three years, more than three years time. [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_5.php クリスチャンルブタン バッグ] Liu Yuxi talks nine were sitting cross-legged cry in silence, Qin Yu himself sitting cross-legged in the room heard in silence.<br><br>retreat at the child in the Qin Yu. Suddenly sensed glare to light, when the struggle to open the eyes and yang |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://www.effigies.com/cgi-bin/guestbook/guestbook.cgi http://www.effigies.com/cgi-bin/guestbook/guestbook.cgi]</li> |
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| | <li>[http://www.fuxiaoliming.com/forum.php?mod=viewthread&tid=208121 http://www.fuxiaoliming.com/forum.php?mod=viewthread&tid=208121]</li> |
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| | </ul> |
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| An n-class association scheme consists of a [[Set (mathematics)|set]] ''X'' together with a [[partition of a set|partition]] ''S'' of ''X'' × ''X'' into n + 1 [[binary relation]]s, R<sub>0</sub>, R<sub>1</sub>, ..., R<sub>n</sub> which satisfy:
| | == leading to a lot of other galaxies Chuansong Zhen. == |
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| *<math>R_{0}=\{(x,x):x\in X\}</math> and is called the [[Identity relation]].
| | Seeking heart.<br><br>'Yu Huang Majesty broad-minded, will not ......'<br><br>see Emperor Xian Yu also persuade the Vatican, Qin Yu busy hand nonchalantly said: 'Well, I do not have time and クリスチャンルブタン 店舗 you nonsense, leave' Qin Yu finished the whole person speed channeling out, after a great diversion again disappeared.<br><br>Qin Yu next destination 'blue star.'<br><br>blue star, as Angelica galactic core planet, this planet in Star Chuansong Zhen course very much, traffic developed, leading to a lot of other galaxies Chuansong Zhen.<br>'Friends dark astral ......'<br>Emperor Xian Yu Vatican wants to follow to catch up.<br><br>'seems only to let クリスチャンルブタン 偽物 Yu Huang personally resolved, I do not know where now arrived.' Bai star in his own to start a summons to Yu Huang, unfortunately there is no communications success.<br><br>Yu クリスチャンルブタン メンズ 靴 Wong and because he is too far away, only to be summoned クリスチャンルブタン 偽物 within the 'maze Communications' success. Vatican last Emperor Xian Yu Lian cents or so ice will matter to Yu Huang Di reporting.<br><br>Emperor Xian Lian from クリスチャンルブタン通販 the ice to his arraignment |
| *Defining <math> R^* :=\{(x,y) | (y,x)\in R\}</math>, if ''R'' in ''S'', then ''R*'' in ''S''
| | 相关的主题文章： |
| *If <math>(x,y)\in R_{k}</math>, the number of <math>z\in X</math> such that <math>(x,z)\in R_{i}</math> and <math>(z,y)\in R_{j}</math> is a constant <math>p^k_{ij}</math> depending on <math>i</math>, <math>j</math>, <math>k</math> but not on the particular choice of <math>x</math> and <math>y</math>.
| | <ul> |
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| | <li>?aid=105609</li> |
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| | <li>?aid=2</li> |
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| | <li>?aid=393439</li> |
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| | </ul> |
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| An association scheme is ''commutative'' if <math>p_{ij}^k=p_{ji}^k</math> for all <math>i</math>, <math>j</math> and <math>k</math>. Most authors assume this property.
| | == palace courtyard alone among the Qin Yu == |
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| A ''symmetric'' association scheme is one in which each relation <math>R_i</math> is a [[symmetric relation]]. That is:
| | Had lost a war and key central repair, while wind figs, QIN talk about the problem of comprehension aspects. As for the injury itself, Qin Yu, has long been [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_8.php クリスチャンルブタン 店舗 東京] a popular tears repaired.<br><br>palace courtyard alone among the Qin [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_10.php クリスチャンルブタン 値段] Yu, Yu Qin, Qin Feng, Qin Zheng, QIN, wind figs five people sitting around the stone table, at the moment on the stone table stood three Cheats, it had left Qin [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_4.php クリスチャンルブタン セール] Yu three comprehension Cheats.<br><br>'Kohane, these three cheats you, you and I have seen the wind uncle, you wind uncle say, this three Cheats comprehension of each of the circles are uppermost comprehension Cheats, you wind uncle are now discarded Over the past practice of law, the use of this method of the above first comprehension. 'QIN smile.<br><br>wind figs also exclaimed: 'Kohane is the original Munekado secret pass to [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン 東京] practice law, but I also know the secret of Yuan Ying gas only, but obviously not as good as these cheats so deep so proud of mysterious practice of law, it is estimated. Munekado town than we have severe cases of canon. '<br><br>Qin Yu [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_0.php クリスチャンルブタン 店舗] is not talking, but with one hand stretched, spiritual goods is five knives and a black rifles floating |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://www.crowningtech.com/plus/feedback.php?aid=251 http://www.crowningtech.com/plus/feedback.php?aid=251]</li> |
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| | <li>[http://hhtea.com/plus/feedback.php?aid=1321 http://hhtea.com/plus/feedback.php?aid=1321]</li> |
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| | <li>[http://www2s.biglobe.ne.jp/~ton/cgbbs/joyful.cgi http://www2s.biglobe.ne.jp/~ton/cgbbs/joyful.cgi]</li> |
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| | </ul> |
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| * if (''x'',''y'') ∈ ''R''<sub>''i''</sub>, then (''y'',''x'') ∈ ''R''<sub>''i''</sub> . (Or equivalently, ''R''* = ''R''.)
| | == 'Nothing == |
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| Every symmetric association scheme is commutative.
| | Other tribe still mulberry family palace. '<br><br>'Nothing, in the other direction, I have arranged the tribe.'<br><br>Sang Mo exposed face of evil spirits, while red robe yixian moment, the people turning to red giant octopus, eight Rouzhu startled, of<br><br>towards the south for a streamer will recover the past, Qin Yu faster than the one who is faster on the trace.<br><br>......<br><br>are flying speed towards the south [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_4.php クリスチャンルブタン 店舗] Qin [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_12.php クリスチャンルブタンジャパン] Yu suddenly stopped.<br><br>'someone!'<br><br>his spiritual knowledge has clearly found dozens of miles away octopus dozen family tribe, although now found only a dozen, but who knows<br><br>if there are more. Sang Mo bound to have come directly toward their location, Qin Yu had a judge.<br><br>'That Sang ink know where I am, but also the formation of encirclement.'<br><br>stopped for a moment in the Qin Yu Qin Yu spiritual knowledge found behind dozens of miles away [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_4.php クリスチャンルブタン パンプス] to have a huge red giant [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_2.php クリスチャンルブタン 東京] octopus came the [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_14.php クリスチャンルブタン 銀座] claws. |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://zhonglaonianfuzhuang.com/plus/feedback.php?aid=8 http://zhonglaonianfuzhuang.com/plus/feedback.php?aid=8]</li> |
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| | <li>[http://www.cuisette.com/index.php?option=com_community&view=profile&userid=96507 http://www.cuisette.com/index.php?option=com_community&view=profile&userid=96507]</li> |
| | |
| | <li>[http://www.yaoyin.net/plus/feedback.php?aid=22 http://www.yaoyin.net/plus/feedback.php?aid=22]</li> |
| | |
| | </ul> |
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| Note, however, that while the notion of an association scheme generalizes the notion of a group, the notion of a commutative association scheme only generalizes the notion of a commutative group.
| | == ' Octagon Chinese == |
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| Two points ''x'' and ''y'' are called ''i'' <sup>th</sup> associates if <math>(x,y)\in R_{i}</math>. The definition states that if ''x'' and ''y'' are ''i'' <sup>th</sup> associates so are ''y'' and ''x''. Every pair of points are ''i'' <sup>th</sup> associates for exactly one <math>i</math>. Each point is its own zeroth associate while distinct points are never zeroth associates. If ''x'' and ''y'' are ''k'' <sup>th</sup> associates then the number of points <math>z</math> which are both ''i'' <sup>th</sup> associates of <math>x</math> and ''j'' <sup>th</sup> associates of <math>y</math> is a constant <math>p^k_{ij}</math>.
| | Have not even touched, really is weird enough.<br><br>......<br><br>forward and move on.<br><br>line of nine people, no one spoke, everyone focus forward, the sky wind density was significantly [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_13.php クリスチャンルブタン パンプス] large knife a lot, even though the number of dead [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_12.php クリスチャンルブタン 値段] and black feather is not enough relaxed.<br><br>and reckless purple pupil is carefully [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_12.php クリスチャンルブタン 値段] look in [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_13.php クリスチャンルブタン通販] all directions.<br><br>suddenly - reckless pupil purple light Dasheng, shot hundreds of feet of purple, go directly toward the distant [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_14.php クリスチャンルブタン 銀座] shot. And in that direction are a group of equally tough vague figure advance.<br><br>'Du Zhongjun!' Octagon Chinese<br><br>foolhardy soon Bao Meng, this sudden loud shout Qin Yu, who not only shocked, even distant Ao Feng, Hua Yan, such as a group of people is also a surprise.<br><br>'is reckless!' Du Zhongjun looked back and his face suddenly became pale.<br><br>'how they come?' Hua Yan, Ao Feng, who look incredible.<br><br>'ah!' distraction months of the empress even touch |
| | | 相关的主题文章： |
| ===Graph interpretation and adjacency matrices===
| | <ul> |
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| An association scheme can be visualized as a [[complete graph]] with labeled edges. The graph has <math>v</math> vertices, one for each point of <math>X</math>, and the edge joining vertices <math>x</math> and <math>y</math> is labeled <math>i</math> if <math>x</math> and <math>y</math> are <math>i</math> <sup>th</sup> associates. Each edge has a unique label, and the number of triangles with a fixed base labeled <math>k</math> having the other edges labeled <math>i</math> and <math>j</math> is a constant <math>p^k_{ij}</math>, depending on <math>i,j,k</math> but not on the choice of the base. In particular, each vertex is incident with exactly <math>p^0_{ii}=v_{i}</math> edges labeled <math>i</math>; <math>v_{i}</math> is the [[Adjacency relation|valency]] of the [[Relation (mathematics)|relation]] <math>R_{i}</math>. There are also loops labeled <math>0</math> at each vertex <math>x</math>, corresponding to <math>R_{0}</math>.
| | <li>[http://www18.tok2.com/home2/syun10/joyful/joyful.cgi http://www18.tok2.com/home2/syun10/joyful/joyful.cgi]</li> |
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| The [[Relation (mathematics)|relations]] are described by their [[Adjacency matrix|adjacency matrices]]. <math>A_{i}</math> is the [[adjacency matrix]] of <math>R_{i}</math> for <math>i=0,\ldots,n</math> and is a ''v'' × ''v'' [[Matrix (mathematics)|matrix]] with rows and columns labeled by the points of <math>X</math>.
| | <li>[http://www.huakungroup.net/bbs/home.php?mod=space&uid=277212 http://www.huakungroup.net/bbs/home.php?mod=space&uid=277212]</li> |
| :<math>\left(A_{i}\right)_{x,y}=\left\{\begin{matrix} | | |
| 1, & \mbox{if } \left(x,y\right)\in R_{i},\\
| | <li>[http://www.maxonidea.com/plus/feedback.php?aid=1 http://www.maxonidea.com/plus/feedback.php?aid=1]</li> |
| 0, & \mbox{otherwise.} \end{matrix}\right. \qquad(1)</math>
| | |
| | | </ul> |
| The definition of an association scheme is equivalent to saying that the <math>A_{i}</math> are ''v'' × ''v'' (0,1)-[[Matrix (mathematics)|matrices]] which satisfy
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| :I. <math>A_{i} \,</math> is symmetric,
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| :II. <math>\sum_{i=0}^{n}A_{i}=J</math> (the all-ones matrix),
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| :III. <math>A_{0}=I \,</math>,
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| :IV. <math>A_{i}A_{j}=\sum_{k=0}^{n} p^k_{ij}A_{k}=A_{j}A_{i}, i,j=0,\ldots,n</math>.
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| The (''x'', ''y'')-th entry of the left side of (IV) is the number of paths of length two between ''x'' and ''y'' with labels i and j in the graph. Note that the rows and columns of <math>A_{i}</math> contain <math>v_{i}</math> <math>1</math>'s:
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| <math>A_{i} J=J A_{i}=v_{i} J. \qquad(2)</math>
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| ===Terminology===
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| *The numbers <math>p_{ij}^k</math> are called the ''parameters'' of the scheme. They are also referred to as the ''structural constants''.
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| ==History==
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| The term ''association scheme'' is due to {{harv|Bose|Shimamoto|1952}} but the concept is already inherent in {{harv|Bose|Nair|1939}}.<ref>{{harvnb|Dembowski|1968|loc=pg. 281, footnote 1}}</ref> These authors were studying what statisticians have called ''partially balanced incomplete block designs'' (PBIBDs). The subject became an object of algebraic interest with the publication of {{harv|Bose|Mesner|1959}} and the introduction of the Bose–Mesner algebra. The most important contribution to the theory was the thesis of P. Delsarte {{harv|Delsarte|1973}} who recognized and fully used the connections with coding theory and design theory.<ref>{{harvnb|Bannai|Ito|1984|loc=pg. vii}}</ref> Generalizations have been studied by D. G. Higman (coherent configurations) and B. Weisfeiler (distance regular graphs).
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| ==Basic facts==
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| *<math>p_{00}^0 = 1</math>, i.e. if <math>(x,y) \in R_0</math> then <math>x = y</math> and the only <math>z</math> such that <math>(x,z) \in R_0</math> is <math>z=x</math>
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| *<math>\sum_{i=0}^{k} p_{ii}^0 = |X|</math>, this is because the <math>R_i</math> partition <math>X</math>.
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| ==The Bose–Mesner algebra==
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| The [[Adjacency matrix|adjacency matrices]] <math>A_i</math> of the [[Graph (mathematics)|graphs]] <math>\left(X,R_{i}\right)</math> generate a [[Commutativity|commutative]] and [[associative]] [[algebra]] <math>\mathcal{A}</math> (over the real or [[complex number]]s) both for the [[matrix product]] and the [[pointwise product]]. This [[Associative algebra|associative]], [[commutative algebra]] is called the [[Bose–Mesner algebra]] of the association scheme.
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| Since the [[Matrix (mathematics)|matrices]] in <math>\mathcal{A}</math> are [[Symmetric matrix|symmetric]] and [[Commutativity|commute]] with each other, they can be [[Diagonal matrix|diagonalized]] simultaneously. Therefore <math>\mathcal{A}</math> is [[Semisimple operator|semi-simple]] and has a unique basis of primitive [[idempotent]]s <math>J_{0},\ldots,J_{n}</math>.
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| There is another [[algebra]] of <math>\left(n+1\right)\times\left(n+1\right)</math> [[Matrix (mathematics)|matrices]] which is [[isomorphic]] to <math>\mathcal{A}</math>, and is often easier to work with.
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| ==Examples==
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| *The [[Johnson scheme]], denoted ''J''(''v,k''), is defined as follows. Let ''S'' be a set with ''v'' elements. The points of the scheme ''J''(''v'',''k'') are the <math>{v \choose k}</math> subsets of S with ''k'' elements. Two ''k''-element subsets ''A'', ''B'' of ''S'' are ''i'' <sup>th</sup> associates when their intersection has size ''k'' − ''i''.
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| *The [[Hamming scheme]], denoted ''H''(''n'',''q''), is defined as follows. The points of ''H''(''n'',''q'') are the ''q<sup>n</sup>'' ordered ''n''-[[tuple]]s over a set of size ''q''. Two ''n''-tuples ''x'', ''y'' are said to be ''i'' <sup>th</sup> associates if they disagree in exactly ''i'' coordinates. E.g., if ''x'' = (1,0,1,1), ''y'' = (1,1,1,1), ''z'' = (0,0,1,1), then ''x'' and ''y'' are 1st associates, ''x'' and ''z'' are 1st associates and ''y'' and ''z'' are 2nd associates in ''H(4,2)''.
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| *A [[distance-regular graph]], ''G'', forms an association scheme by defining two vertices to be ''i'' <sup>th</sup> associates if their distance is ''i''.
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| *A [[finite group]] ''G'' yields an association scheme on <math>X=G</math>, with a class ''R''<sub>''g''</sub> for each group element, as follows: for each <math>g \in G</math> let <math>R_g=\{(x,y) | x=g*y\}</math> where <math>*</math> is the group [[operation (mathematics)|operation]]. The class of the group identity is ''R''<sub>0</sub>. This association scheme is commutative if and only if ''G'' is [[Abelian group|abelian]].
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| *A specific 3-class association scheme:<ref>{{harvnb|Street|Street|1987|loc=pg. 238}}</ref>
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| :Let ''A''(3) be the following association scheme with three associate classes on the set ''X'' = {1,2,3,4,5,6}. The (''i'',''j'') entry is ''s'' if elements ''i'' and ''j'' are in relation R<sub>s</sub>.
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| <center>
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| {| class="wikitable"
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| |-
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| ! !! 1!! 2!! 3!! 4!! 5!! 6
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| |-
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| | '''1''' || <span style="color:white; background:blue"> 0 </span> || <span style="color:white; background:red"> 1 </span> || <span style="color:white; background:red"> 1 </span> || <span style="color:white; background:lime"> 2 </span> || <span style="color:white; background:fuchsia"> 3 </span> || <span style="color:white; background:fuchsia"> 3 </span>
| |
| |-
| |
| |'''2'''|| <span style="color:white; background:red"> 1 </span> || <span style="color:white; background:blue"> 0 </span> || <span style="color:white; background:red"> 1 </span> || <span style="color:white; background:fuchsia"> 3 </span> || <span style="color:white; background:lime"> 2 </span> || <span style="color:white; background:fuchsia"> 3 </span>
| |
| |-
| |
| | '''3''' || <span style="color:white; background:red"> 1 </span> || <span style="color:white; background:red"> 1 </span> || <span style="color:white; background:blue"> 0 </span> || <span style="color:white; background:fuchsia"> 3 </span> || <span style="color:white; background:fuchsia"> 3 </span> || <span style="color:white; background:lime"> 2 </span>
| |
| |-
| |
| | '''4''' || <span style="color:white; background:lime"> 2 </span> || <span style="color:white; background:fuchsia"> 3 </span> || <span style="color:white; background:fuchsia"> 3 </span> || <span style="color:white; background:blue"> 0 </span> || <span style="color:white; background:red"> 1 </span> || <span style="color:white; background:red"> 1 </span>
| |
| |-
| |
| | '''5''' || <span style="color:white; background:fuchsia"> 3 </span> || <span style="color:white; background:lime"> 2 </span> || <span style="color:white; background:fuchsia"> 3 </span> || <span style="color:white; background:red"> 1 </span> || <span style="color:white; background:blue"> 0 </span> || <span style="color:white; background:red"> 1 </span>
| |
| |-
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| | '''6''' || <span style="color:white; background:fuchsia"> 3 </span> || <span style="color:white; background:fuchsia"> 3 </span> || <span style="color:white; background:lime"> 2 </span> || <span style="color:white; background:red"> 1 </span> || <span style="color:white; background:red"> 1 </span> || <span style="color:white; background:blue"> 0 </span>
| |
| |}
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| </center> | |
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| ==Coding theory==
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| The [[Hamming scheme]] and the [[Johnson scheme]] are of major significance in classical [[coding theory]].
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| In [[coding theory]], association scheme theory is mainly concerned with the [[Hamming distance|distance]] of a [[code]]. The [[linear programming]] method produces upper bounds for the size of a [[code]] with given minimum [[Hamming distance|distance]], and lower bounds for the size of a [[T-design|design]] with a given strength. The most specific results are obtained in the case where the underlying association scheme satisfies certain [[polynomial]] properties; this leads one into the realm of [[orthogonal polynomials]]. In particular, some universal bounds are derived for [[code]]s and [[T-design|designs]] in polynomial-type association schemes.
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| In classical [[coding theory]], dealing with [[code]]s in a [[Hamming scheme]], the MacWilliams transform involves a family of orthogonal polynomials known as the [[Krawtchouk polynomials]]. These polynomials give the [[eigenvalues]] of the distance relation [[Matrix (mathematics)|matrices]] of the [[Hamming scheme]].
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| ==See also==
| |
| <!-- * [[Analysis of variance]] (Statistics) -->
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| * [[Block design]]
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| * [[Bose–Mesner algebra]]
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| * [[Combinatorial design]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| * {{citation| first=Rosemary A.| last=Bailey | authorlink=Rosemary A. Bailey | url=http://www.maths.qmul.ac.uk/~rab/Asbook | title=Association Schemes: Designed Experiments, Algebra and Combinatorics|publisher=Cambridge University Press|year=2004|isbn=978-0-521-82446-0| mr=2047311}}. (Chapters from preliminary draft are [http://www.maths.qmw.ac.uk/~rab available on-line].)
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| * {{citation| last1=Bannai | first1=Eiichi | last2=Ito | first2=Tatsuro | title=Algebraic combinatorics I: Association schemes | publisher=The Benjamin/Cummings Publishing Co., Inc. | location=Menlo Park, CA | year=1984 | pages=xxiv+425 | isbn=0-8053-0490-8 | mr=0882540 | unused_data=<!-- authorlink1=Eiichi Bannai | authorlink2= Tatsuro Ito --> }}
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| * {{citation| last1=Bose|first1=R. C.|authorlink1=R. C. Bose|first2= K. R.|last2= Nair|title= Partially balanced incomplete block designs|journal= [[Sankhya (journal)|Sankhyā]]|volume= 4|year=1939|pages= 337–372}}
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| * {{citation|last=Bose|first=R. C.|last2=Shimamoto|first2=T.|title=Classification and analysis of partially balanced incomplete block designs with two associate classes|journal=Journal of the American Statistical Association|year=1952|volume=47|pages=151–184|authorlink=R. C. Bose}}
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| * P. Camion (1998), Codes and Association Schemes: Basic Properties of Association Schemes Relevant to Coding, in ''Handbook of Coding Theory'', V. S. Pless and W. C. Huffman, Eds., Elsevier, The Netherlands.
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| {{Experimental design|state=collapsed}}
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| {{Statistics|collection|state=collapsed}}
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| [[Category:Articles with inconsistent citation formats]]
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| [[Category:Design of experiments]]
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| [[Category:Analysis of variance]]
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| [[Category:Algebraic combinatorics]]
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| [[Category:Representation theory]]
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| <!-- [[Category:Algebra]]
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there are lotus.
In a lot of colorful lotus land records.
ordinary barrel. Also half-human Heights barrel. Half a bucket can be.
and this pond. Nearly circular shape. About nearly two meters in diameter, so big pond. Inside the one yuan heavy water. Estimated to be sufficient to store ten barrels of heavy water to one yuan.
both ends of クリスチャンルブタン 靴 メンズ the pond in the クリスチャンルブタン バッグ yuan heavy water. Zhang Lin just two colorful, colorful lotus really look the same as クリスチャンルブタン セール described in the reel. Lotus is divided into 288, lotus minimal, there are lotus.
'Huh?' Qin クリスチャンルブタン 靴 Yu suddenly surprised again.
'central pond is arranged a little to hide the magic circle.' Qin Yu force to explore space discovered. This pond central in the end what is being used to hide the magic array block.
the passage of time, then quickly passed more than three years, more than three years time. クリスチャンルブタン バッグ Liu Yuxi talks nine were sitting cross-legged cry in silence, Qin Yu himself sitting cross-legged in the room heard in silence.
retreat at the child in the Qin Yu. Suddenly sensed glare to light, when the struggle to open the eyes and yang
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leading to a lot of other galaxies Chuansong Zhen.
Seeking heart.
'Yu Huang Majesty broad-minded, will not ......'
see Emperor Xian Yu also persuade the Vatican, Qin Yu busy hand nonchalantly said: 'Well, I do not have time and クリスチャンルブタン 店舗 you nonsense, leave' Qin Yu finished the whole person speed channeling out, after a great diversion again disappeared.
Qin Yu next destination 'blue star.'
blue star, as Angelica galactic core planet, this planet in Star Chuansong Zhen course very much, traffic developed, leading to a lot of other galaxies Chuansong Zhen.
'Friends dark astral ......'
Emperor Xian Yu Vatican wants to follow to catch up.
'seems only to let クリスチャンルブタン 偽物 Yu Huang personally resolved, I do not know where now arrived.' Bai star in his own to start a summons to Yu Huang, unfortunately there is no communications success.
Yu クリスチャンルブタン メンズ 靴 Wong and because he is too far away, only to be summoned クリスチャンルブタン 偽物 within the 'maze Communications' success. Vatican last Emperor Xian Yu Lian cents or so ice will matter to Yu Huang Di reporting.
Emperor Xian Lian from クリスチャンルブタン通販 the ice to his arraignment
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palace courtyard alone among the Qin Yu
Had lost a war and key central repair, while wind figs, QIN talk about the problem of comprehension aspects. As for the injury itself, Qin Yu, has long been クリスチャンルブタン 店舗 東京 a popular tears repaired.
palace courtyard alone among the Qin クリスチャンルブタン 値段 Yu, Yu Qin, Qin Feng, Qin Zheng, QIN, wind figs five people sitting around the stone table, at the moment on the stone table stood three Cheats, it had left Qin クリスチャンルブタン セール Yu three comprehension Cheats.
'Kohane, these three cheats you, you and I have seen the wind uncle, you wind uncle say, this three Cheats comprehension of each of the circles are uppermost comprehension Cheats, you wind uncle are now discarded Over the past practice of law, the use of this method of the above first comprehension. 'QIN smile.
wind figs also exclaimed: 'Kohane is the original Munekado secret pass to クリスチャンルブタン 東京 practice law, but I also know the secret of Yuan Ying gas only, but obviously not as good as these cheats so deep so proud of mysterious practice of law, it is estimated. Munekado town than we have severe cases of canon. '
Qin Yu クリスチャンルブタン 店舗 is not talking, but with one hand stretched, spiritual goods is five knives and a black rifles floating
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'Nothing
Other tribe still mulberry family palace. '
'Nothing, in the other direction, I have arranged the tribe.'
Sang Mo exposed face of evil spirits, while red robe yixian moment, the people turning to red giant octopus, eight Rouzhu startled, of
towards the south for a streamer will recover the past, Qin Yu faster than the one who is faster on the trace.
......
are flying speed towards the south クリスチャンルブタン 店舗 Qin クリスチャンルブタンジャパン Yu suddenly stopped.
'someone!'
his spiritual knowledge has clearly found dozens of miles away octopus dozen family tribe, although now found only a dozen, but who knows
if there are more. Sang Mo bound to have come directly toward their location, Qin Yu had a judge.
'That Sang ink know where I am, but also the formation of encirclement.'
stopped for a moment in the Qin Yu Qin Yu spiritual knowledge found behind dozens of miles away クリスチャンルブタン パンプス to have a huge red giant クリスチャンルブタン 東京 octopus came the クリスチャンルブタン 銀座 claws.
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' Octagon Chinese
Have not even touched, really is weird enough.
......
forward and move on.
line of nine people, no one spoke, everyone focus forward, the sky wind density was significantly クリスチャンルブタン パンプス large knife a lot, even though the number of dead クリスチャンルブタン 値段 and black feather is not enough relaxed.
and reckless purple pupil is carefully クリスチャンルブタン 値段 look in クリスチャンルブタン通販 all directions.
suddenly - reckless pupil purple light Dasheng, shot hundreds of feet of purple, go directly toward the distant クリスチャンルブタン 銀座 shot. And in that direction are a group of equally tough vague figure advance.
'Du Zhongjun!' Octagon Chinese
foolhardy soon Bao Meng, this sudden loud shout Qin Yu, who not only shocked, even distant Ao Feng, Hua Yan, such as a group of people is also a surprise.
'is reckless!' Du Zhongjun looked back and his face suddenly became pale.
'how they come?' Hua Yan, Ao Feng, who look incredible.
'ah!' distraction months of the empress even touch
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