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| In [[mathematics]], the '''Leech lattice''' is an even [[unimodular lattice]] Λ<sub>24</sub> in 24-dimensional [[Euclidean space]] '''E'''<sup>24</sup> found by {{harvs|txt|authorlink=John Leech (mathematician)|first=John|last= Leech|year=1967}}.
| | Over time, the data on a difficult drive gets scattered. Defragmenting your hard drive puts the data back into sequential purchase, making it simpler for Windows to access it. As a result, the performance of the computer usually enhance. An excellent registry cleaner allows work this task. However in the event you would like to defrag the PC with Windows software. Here a link to show we how.<br><br>Another answer will be to supply the computer system with a new msvcr71 file. Frequently, once the file has been corrupted or damaged, it may no longer be able to function like it did before so it's only natural to substitute the file. Just download another msvcr71.dll file within the internet. Frequently, the file comes inside a zip format. Extract the files from the zip folder and spot them accordingly inside this location: C:\Windows\System32. Afterward, re-register the file. Click Start and then choose Run. When the Run window appears, kind "cmd". Press Enter plus then kind "regsvr32 -u msvcr71.dll" followed by "regsvr32 msvcr71.dll". Press Enter again plus the file could be registered accordingly.<br><br>So, this advanced dual scan is not just among the better, nevertheless it really is furthermore freeware. And as of all of this which numerous regard CCleaner one of the better registry products in the market now. I would add that I personally like Regcure for the easy reason which it has a greater interface and I understand for a fact it is ad-ware without charge.<br><br>Check your Windows taskbar, which is found on the lower proper hand corner of your computer screen. This taskbar consist of programs you have running inside the background. If you have too several of them, they usually take your computer's resources.<br><br>After which, I also bought the Regtool [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities] Software, and it further protected my computer having system crashes. All my registry problems are fixed, plus I will function peacefully.<br><br>Why this issue arises frequently? What are the causes of it? In fact, there are 3 major causes which can lead to the PC freezing problem. To resolve the problem, we have to take 3 procedures in the following paragraphs.<br><br>You want an option to automatically delete unwanted registry keys. This usually help save you hours of laborious checking by your registry keys. Automatic deletion is a key element whenever you compare registry cleaners.<br><br>Another significant program you'll wish To receive is a registry cleaner. The registry is a big list of everything installed on the computer, plus Windows references it when it opens a program or uses a device connected to the computer. Whenever you delete a program, its registry entry could moreover be deleted, however, sometimes it's not. A registry cleaner will do away with these older entries so Windows could search the registry faster. It additionally deletes or corrects any entries that viruses have corrupted. |
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| ==History==
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| Many of the cross-sections of the Leech lattice, including the [[Coxeter–Todd lattice]] and [[Barnes–Wall lattice]], in 12 and 16 dimensions, were found much earlier than the Leech lattice. {{harvtxt|O'Connor|Pall|1944}} discovered a related odd unimodular lattice in 24 dimensions, now called the odd Leech lattice, whose even sublattice has index 2 in the Leech lattice. The Leech lattice was discovered in 1965 by {{harvs|txt|authorlink=John Leech (mathematician)|first=John|last= Leech|year=1967|loc=2.31, p. 262}}, by improving some earlier sphere packings he found {{harv|Leech|1964}}.
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| {{harvs|txt|authorlink=John Horton Conway|last=Conway|year=1968}} calculated the order of the automorphism group of the Leech lattice, and, working with [[John G. Thompson]], discovered three new [[sporadic group]]s as a by-product: the [[Conway groups]], Co<sub>1</sub>, Co<sub>2</sub>, Co<sub>3</sub>. They also rediscovered four other (then) recently announced sporadic groups, namely, [[Higman-Sims group|Higman-Sims]], [[Suzuki sporadic group|Suzuki]], [[McLaughlin group (mathematics)|McLaughlin]], and the [[Janko group]] J<sub>2</sub> (Ronan, p.155)
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| {{quote box
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| |align=right
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| |quote=Bei dem Versuch, eine Form aus einer solchen Klasse wirklich anzugeben, fand ich mehr als 10 verschiedene Klassen in Γ<sub>24</sub>
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| |source={{harvtxt|Witt|1941|loc=p.324}}
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| }}
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| In a seminar in 1970 [[Ernst Witt]] claimed that one of the lattices he found in 1940 was the Leech lattice. {{harvtxt|Witt|1941|loc=p.324}}, has a single rather cryptic sentence mentioning that he found more than 10 even unimodular lattices in 24 dimensions without giving further details. See his collected works {{harv|Witt|1998|loc=p. 328–329}} for more comments and for some notes Witt wrote about this in 1972.
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| ==Characterization==
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| The Leech lattice Λ<sub>24</sub> is the unique lattice in '''E'''<sup>24</sup> with the following list of properties:
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| *It is [[unimodular lattice|unimodular]]; i.e., it can be generated by the columns of a certain 24×24 [[matrix (mathematics)|matrix]] with [[determinant]] 1.
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| *It is even; i.e., the square of the length of any vector in Λ<sub>24</sub> is an even integer.
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| *The length of any non-zero vector in Λ<sub>24</sub> is at least 2.
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| ==Properties==
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| The last condition is equivalent to the condition that unit balls centered at the points of Λ<sub>24</sub> do not overlap. Each is tangent to 196,560 neighbors, and this is known to be the largest number of non-overlapping 24-dimensional unit balls that can simultaneously touch a single unit ball (compare with 6 in dimension 2, as the maximum number of pennies which can touch a central penny; see [[kissing number]]). This arrangement of 196560 unit balls centred about another unit ball is so efficient that there is no room to move any of the balls; this configuration, together with its mirror-image, is the ''only'' 24-dimensional arrangement where 196560 unit balls simultaneously touch another. This property is also true in 1, 2 and 8 dimensions, with 2, 6 and 240 unit balls, respectively, based on the [[integer lattice]], [[hexagonal tiling]] and [[E8 lattice]], respectively.
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| It has no [[root system]] and in fact is the first [[unimodular lattice]] with no ''roots'' (vectors of norm less than 4), and therefore has a centre density of 1. By multiplying this value by the volume of a unit ball in 24 dimensions, <math>\tfrac{\pi^{12}}{12!}</math>, one can derive its absolute density.
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| {{harvtxt|Conway|1983}} showed that the Leech lattice is isometric to the set of simple roots (or the [[Dynkin diagram]]) of the [[reflection group]] of the 26-dimensional even Lorentzian unimodular lattice [[II25,1|II<sub>25,1</sub>]]. By comparison, the Dynkin diagrams of II<sub>9,1</sub> and II<sub>17,1</sub> are finite.
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| ==Constructions==
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| The Leech lattice can be constructed in a variety of ways. As with all lattices, it can be constructed via its [[generator matrix]], a 24×24 matrix with [[determinant]] 1.
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| ===Using the binary Golay code===
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| The Leech lattice can be explicitly constructed as the set of vectors of the form 2<sup>−3/2</sup>(''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>24</sub>) where the ''a<sub>i</sub>'' are integers such that
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| :<math>a_1+a_2+\cdots+a_{24}\equiv 4a_1\equiv 4a_2\equiv\cdots\equiv4a_{24}\pmod{8}</math>
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| and for each fixed residue class modulo 4, the 24 bit word, whose 1's correspond to the coordinates ''i'' such that ''a''<sub>i</sub> belongs to this residue class, is a word in the [[binary Golay code]]. The Golay code, together with the related Witt Design, features in a construction for the 196560 minimal vectors in the Leech lattice.
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| ===Using the Lorentzian lattice II<sub>25,1</sub>===
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| The Leech lattice can also be constructed as <math>w^\perp/w</math> where ''w'' is the Weyl vector:
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| :<math>(0,1,2,3,\dots,22,23,24; 70)</math> | |
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| in the 26-dimensional even Lorentzian [[unimodular lattice]] [[II25,1|II<sub>25,1</sub>]]. The existence of such an integral vector of norm zero relies on the fact that 1<sup>2</sup> + 2<sup>2</sup> + ... + 24<sup>2</sup> is a [[square number|perfect square]] (in fact 70<sup>2</sup>); the [[number 24]] is the only integer bigger than 1 with this property. This was conjectured by [[Édouard Lucas]], but the proof came much later, based on [[elliptic functions]].
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| The vector
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| <math>(0,1,2,3,\dots,22,23,24)</math> | |
| in this construction is really the [[Weyl vector]] of the even sublattice ''D''<sub>24</sub> of the odd unimodular lattice ''I''<sup>25</sup>. More generally, if ''L'' is any positive definite unimodular lattice of dimension 25 with at least 4 vectors of norm 1, then the Weyl vector of its norm 2 roots has integral length, and there is a similar construction of the Leech lattice using ''L'' and this Weyl vector.
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| ===Based on other lattices===
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| {{harvtxt|Conway|Sloane|1982}} described another 23 constructions for the Leech lattice, each based on a [[Niemeier lattice]]. It can also be constructed by using three copies of the [[E8 lattice]], in the same way that the binary Golay code can be constructed using three copies of the extended [[Hamming code]], H<sub>8</sub>. This construction is known as the '''Turyn''' construction of the Leech lattice.
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| ===As a laminated lattice===
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| Starting with a single point, Λ<sub>0</sub>, one can stack copies of the lattice Λ<sub>n</sub> to form an (''n'' + 1)-dimensional lattice, Λ<sub>''n''+1</sub>, without reducing the minimal distance between points. Λ<sub>1</sub> corresponds to the [[integer lattice]], Λ<sub>2</sub> is to the [[hexagonal lattice]], and Λ<sub>3</sub> is the [[face-centered cubic]] packing. {{harvtxt|Conway|Sloane|1982b}} showed that the Leech lattice is the unique laminated lattice in 24 dimensions.
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| ===As a complex lattice===
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| The Leech lattice is also a 12-dimensional lattice over the [[Eisenstein integers]]. This is known as the '''complex Leech lattice''', and is isomorphic to the 24-dimensional real Leech lattice. In the complex construction of the Leech lattice, the [[binary Golay code]] is replaced with the [[ternary Golay code]]{{clarify|date=June 2012}}, and the Mathieu group ''M''<sub>24</sub> is replaced with the Mathieu group ''M''<sub>12</sub>{{clarify|date=June 2012}}. The ''E''<sub>6</sub> lattice, ''E''<sub>8</sub> lattice and [[Coxeter–Todd lattice]] also have constructions as complex lattices, over either the Eisenstein or [[Gaussian integers]].
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| ===Using the icosian ring===
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| The Leech lattice can also be constructed using the ring of [[icosian]]s. The icosian ring is abstractly isomorphic to the [[E8 lattice]], three copies of which can be used to construct the Leech lattice using the Turyn construction.
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| ==Symmetries==
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| The Leech lattice is highly symmetrical. Its [[automorphism group]] is the double cover of the [[Conway group]] Co<sub>1</sub>; its order is 8 315 553 613 086 720 000. Many other [[sporadic simple group]]s, such as the remaining Conway groups and [[Mathieu groups]], can be constructed as the stabilizers of various configurations of vectors in the Leech lattice.
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| Despite having such a high ''rotational'' symmetry group, the Leech lattice does not possess any lines of reflection symmetry. In other words, the Leech lattice is [[chiral]].
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| The automorphism group was first described by [[John Horton Conway|John Conway]]. The 398034000 vectors of norm 8 fall into 8292375 'crosses' of 48 vectors. Each cross contains 24 mutually orthogonal vectors and their inverses, and thus describe the vertices of a 24-dimensional [[orthoplex]]. Each of these crosses can be taken to be the coordinate system of the lattice, and has the same symmetry of the [[Golay code]], namely 2<sup>12</sup> × |M<sub>24</sub>|. Hence the full automorphism group of the Leech lattice has order 8292375 × 4096 × 244823040, or 8 315 553 613 086 720 000.
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| ==Geometry==
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| {{harvtxt|Conway|Parker|Sloane|1982}} showed that the covering radius of the Leech lattice is <math>\sqrt 2</math>; in other words, if we put a closed ball of this radius around each lattice point, then these just cover Euclidean space. The points at distance at least <math>\sqrt 2</math> from all lattice points are called the '''''deep holes''''' of the Leech lattice. There are 23 orbits of them under the automorphism group of the Leech lattice, and these orbits correspond to the 23 [[Niemeier lattices]] other than the Leech lattice: the set of vertices of deep hole is isometric to the affine Dynkin diagram of the corresponding Niemeier lattice.
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| The Leech lattice has a density of <math>\tfrac{\pi^{12}}{12!}\approx 0.001930</math>, correct to six decimal places. Cohn and Kumar showed that it gives the densest ''lattice'' [[sphere packing|packing of balls]] in 24-dimensional space. Their results suggest, but do not prove, that this configuration also gives the densest among all packings of balls in 24-dimensional space. No arrangement of 24-dimensional spheres can be denser than the Leech lattice by a factor of more than 1.65×10<sup>−30</sup>, and it is highly probable that the Leech lattice is indeed optimal.
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| The 196560 minimal vectors are of three different varieties, known as ''shapes'':
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| * 1104 vectors of shape (4<sup>2</sup>,0<sup>22</sup>), for all permutations and sign choices;
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| * 97152 vectors of shape (2<sup>8</sup>,0<sup>16</sup>), where the '2's correspond to octads in the Golay code, and there an even number of minus signs;
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| * 98304 vectors of shape (3,1<sup>23</sup>), where the signs come from the Golay code, and the '3' can appear in any position.
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| The [[ternary Golay code]], [[binary Golay code]] and Leech lattice give very efficient 24-dimensional [[spherical code]]s of 729, 4096 and 196560 points, respectively. Spherical codes are higher-dimensional analogues of [[Tammes problem]], which arose as an attempt to explain the distribution of pores on pollen grains. These are distributed as to maximise the minimal angle between them. In two dimensions, the problem is trivial, but in three dimensions and higher it is not. An example of a spherical code in three dimensions is the set of the 12 vertices of the regular icosahedron.
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| ==Theta series==
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| One can associate to any (positive-definite) lattice Λ a [[theta function]] given by
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| :<math>\Theta_\Lambda(\tau) = \sum_{x\in\Lambda}e^{i\pi\tau\|x\|^2}\qquad\mathrm{Im}\,\tau > 0.</math>
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| The theta function of a lattice is then a [[holomorphic function]] on the [[upper half-plane]]. Furthermore, the theta function of an even unimodular lattice of rank ''n'' is actually a [[modular form]] of weight ''n''/2. The theta function of an integral lattice is often written as a power series in <math>q = e^{2i\pi\tau}</math> so that the coefficient of ''q''<sup>''n''</sup> gives the number of lattice vectors of norm 2''n''. In the Leech lattice, there are 196560 vectors of norm 4, 16773120 vectors of norm 6, 398034000 vectors of norm 8 and so on. The theta series of the Leech lattice is thus:
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| : <math>\sum_{m=0}^\infty \frac{65520}{691}\left(\sigma_{11} (m) - \tau (m) \right) q^{m} = 1 + 196560q^2 + 16773120q^3 + 398034000q^4 + \cdots</math>
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| where <math>\tau(n)</math> represents the [[Ramanujan tau function]], and <math>\sigma_{11}(n)</math> is a [[divisor function]]. It follows that the number of vectors of norm 2''m'' is
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| : <math> \frac{65520}{691} \left(\sigma_{11} (m) - \tau (m) \right).</math>
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| ==Applications==
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| The [[vertex algebra]] of the [[conformal field theory]] describing [[bosonic string theory]], compactified on the 24-dimensional [[quotient group|quotient]] [[torus]] '''R'''<sup>24</sup>/Λ<sub>24</sub> and [[orbifold]]ed by a two-element reflection group, provides an explicit construction of the [[Griess algebra]] that has the [[monster group]] as its automorphism group. This '''[[monster vertex algebra]]''' was also used to prove the [[monstrous moonshine]] conjectures.
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| The [[binary Golay code]], independently developed in 1949, is an application in [[coding theory]]. More specifically, it is an error-correcting code capable of correcting up to three errors in each 24-bit word, and detecting a fourth. It was used to communicate with the [[Voyager probes]], as it is much more compact than the previously-used [[Hadamard code]].
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| [[Quantizer]]s, or [[analog-to-digital converter]]s, can use lattices to minimise the average [[root-mean-square]] error. Most quantizers are based on the one-dimensional [[integer lattice]], but using multi-dimensional lattices reduces the RMS error. The Leech lattice is a good solution to this problem, as the [[Voronoi cell]]s have a low [[second moment]].
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| ==See also==
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| *[[Conway group]]
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| *[[Sphere packing]]
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| *[[E8 lattice|E<sub>8</sub> lattice]]
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| ==References==
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| * {{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | title=A perfect group of order 8,315,553,613,086,720,000 and the sporadic simple groups | mr=0237634 | year=1968 | journal=[[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | volume=61 | pages=398–400 | doi=10.1073/pnas.61.2.398 | issue=2}}
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| *{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | title=The automorphism group of the 26-dimensional even unimodular Lorentzian lattice | doi=10.1016/0021-8693(83)90025-X | mr=690711 | year=1983 | journal=[[Journal of Algebra]] | issn=0021-8693 | volume=80 | issue=1 | pages=159–163}}
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| *{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Sloane | first2=N. J. A. | author2-link=Neil Sloane | title=Laminated lattices | doi=10.2307/2007025 | mr=678483 | year=1982b | journal=[[Annals of Mathematics|Annals of Mathematics. Second Series]] | issn=0003-486X | volume=116 | issue=3 | pages=593–620}}
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| *{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Parker | first2=R. A. | last3=Sloane | first3=N. J. A. | author3-link=Neil Sloane | title=The covering radius of the Leech lattice | doi=10.1098/rspa.1982.0042 | mr=660415 | year=1982 | journal=[[Proceedings of the Royal Society A]] | issn=0080-4630 | volume=380 | issue=1779 | pages=261–290}}
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| *{{Citation | last1=Conway | first1=John Horton | author1-link=John Horton Conway | last2=Sloane | first2=N. J. A. | author2-link=Neil Sloane | title=Twenty-three constructions for the Leech lattice | doi=10.1098/rspa.1982.0071 | mr=661720 | year=1982 | journal=[[Proceedings of the Royal Society A]] | issn=0080-4630 | volume=381 | issue=1781 | pages=275–283}}
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| *[[John Horton Conway|Conway, J. H.]]; [[Neil Sloane|Sloane, N. J. A.]] (1999). ''Sphere packings, lattices and groups.'' (3rd ed.) With additional contributions by E. Bannai, [[Richard Borcherds|R. E. Borcherds]], [[John Leech (mathematician)|John Leech]], [[Simon P. Norton]], [[Andrew Odlyzko|A. M. Odlyzko]], [[Richard A. Parker]], L. Queen and B. B. Venkov. Grundlehren der Mathematischen Wissenschaften, 290. New York: Springer-Verlag. ISBN 0-387-98585-9.
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| *{{Citation | last1=Leech | first1=John | title=Some sphere packings in higher space | url=http://cms.math.ca/10.4153/CJM-1964-065-1 | doi=10.4153/CJM-1964-065-1 | mr=0167901 | year=1964 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=16 | pages=657–682}}
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| *{{Citation | last1=Leech | first1=John | title=Notes on sphere packings | url=http://cms.math.ca/10.4153/CJM-1967-017-0 | doi=10.4153/CJM-1967-017-0 | mr=0209983 | year=1967 | journal=[[Canadian Journal of Mathematics]] | issn=0008-414X | volume=19 | pages=251–267}}
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| *{{Citation | last1=O'Connor | first1=R. E. | last2=Pall | first2=G. | title=The construction of integral quadratic forms of determinant 1 | doi=10.1215/S0012-7094-44-01127-0 | mr=0010153 | year=1944 | journal=[[Duke Mathematical Journal]] | issn=0012-7094 | volume=11 | pages=319–331}}
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| * Thompson, Thomas M.: "From Error Correcting Codes through Sphere Packings to Simple Groups", Carus Mathematical Monographs, Mathematical Association of America, 1983.
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| *{{Citation | last1=Witt | first1=Ernst | author1-link=Ernst Witt | title=Eine Identität zwischen Modulformen zweiten Grades | doi=10.1007/BF02940750 | mr=0005508 | year=1941 | journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg | volume=14 | pages=323–337}}
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| *{{Citation | last1=Witt | first1=Ernst | author1-link=Ernst Witt | title=Collected papers. Gesammelte Abhandlungen | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-57061-5 | mr=1643949 | year=1998}}
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| * [[Griess]], Robert L.: ''Twelve Sporadic Groups'', Springer-Verlag, 1998.
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| * Ronan, Mark: ''Symmetry and the Monster'', Oxford University Press. ISBN 978-0-19-280723-6.
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| * [[Marcus du Sautoy]]: ''Finding Moonshine''. ISBN 978-0-00-721462-4.
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| ==External links==
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| *[http://cp4space.wordpress.com/2013/09/12/leech-lattice/ Leech lattice (CP4space)]
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| *{{MathWorld|id=LeechLattice}}
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| *[http://www.math.uic.edu/~ronan/Leech_Lattice The Leech Lattice, U. of Illinois at Chicago, Mark Ronan's website]
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| *[http://math.berkeley.edu/~reb/papers/ Papers by R. E. Borcherds]
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| {{DEFAULTSORT:Leech Lattice}}
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| [[Category:Quadratic forms]]
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| [[Category:Lattice points]]
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| [[Category:Sporadic groups]]
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| [[Category:Moonshine theory]]
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Over time, the data on a difficult drive gets scattered. Defragmenting your hard drive puts the data back into sequential purchase, making it simpler for Windows to access it. As a result, the performance of the computer usually enhance. An excellent registry cleaner allows work this task. However in the event you would like to defrag the PC with Windows software. Here a link to show we how.
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