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Topic: Leech lattice

	ARCC Workshop: Sphere Packings, Lattices, and Infinite Dimensional Algebra
		A {\it lattice packing} in ${\Bbb R}^{n}$ is then a sphere packing where the centers of spheres are placed at the points of a lattice $L\subset {\Bbb R}^{n}$, the radius of each sphere being half the length of the shortest non-zero vectors in $L$.
		The Leech lattice appears in several places in `Moonshine' which is a term first coined by J. Conway and S. Norton in 1979 to describe the mysterious connections between finite sporadic simple groups and modular functions.
		An investigation of the techniques of Cohn and Elkies, and of Cohn and Kumar, which are conjectured to give rise to a proof that the $E_8$ root lattice and the Leech lattice give the densest sphere packings in ${\Bbb R}^{8}$ and ${\Bbb R}^{24}$ respectively.
	aimath.org /ARCC/workshops/spherepacking.html (617 words)

	Leech lattice - Wikipedia, the free encyclopedia
		In mathematics, the Leech lattice is a lattice Λ in R
		There are 23 orbits of them, and they correspond to the 23 Niemeier lattices other than the Leech lattice.
		Conway showed that the Leech lattice is isometric to the Dynkin diagram of the reflection group of the 26-dimensional even Lorentzian unimodular lattice II
	en.wikipedia.org /wiki/Leech_lattice (459 words)

	Lattice
		Every lattice can be generated from a basis for the underlying vector space by considering all linear combinations with integral coefficients.
		In another mathematical usage, a lattice is a partially ordered set in which all nonempty finite subsets have a least upper bound and a greatest lower bound (also called supremum and infimum, respectively).
		The lattice of submodules of a module and the lattice of normal subgroups of a group have the special property that x v (y ^ (x v z)) = (x v y) ^ (x v z) for all x, y and z in the lattice.
	www.ebroadcast.com.au /lookup/encyclopedia/la/Lattice.html (913 words)

	The Leech Lattice (Site not responding. Last check: 2007-10-22)
		It depicts a portion of the 24-dimensional Leech lattice.
		The 1 brown node is the next closest lattice point, followed by the 3 red nodes, the 6 orange nodes, and the 10 green nodes.
		A typical 2-dimensional lattice is given by the vertices (or the centers) of a tiling by square tiles.
	faculty.ccp.edu /dept/math/math_logo.html (359 words)

	Leech biography
		Leech is, however, best known for the Leech lattice which gives rise to three sporadic
		Leech knew that the symmetry group would be interesting, and he worked on it for some time giving a lower bound for its order (which later proved to be the actual order of the group).
		Leech died almost exactly one month after Gorenstein who had overseen the classification of finite simple groups.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Leech.html (649 words)

	4-dim HyperDiamond Lattice
		The 8-dimensional HyperDiamond octonionic E8 lattice is associated with the octonionic X-product of Cederwall and Preitschopf and a later paper of Dixon.
		The /\16 lattice is associated with the octonionic XY-product of Dixon.
		Since the D4 lattice is the lattice of integral quaternions, the even D4 of D4+ is the integral quaternion lattice expanded by a factor of 2 so that each layer is twice as far from the origin (and, in particular, the closest layer is at distance 2 instead of 1 from the origin).
	www.valdostamuseum.org /hamsmith/FynCkb.html (6224 words)

	Lattice - Wikipedia, the free encyclopedia
		Look up lattice in Wiktionary, the free dictionary.
		Bravais lattice, 14 possible arrangements of repeating points in 3-D
		Lattice model (physics), a model defined not on a continuum, but on a lattice
	en.wikipedia.org /wiki/Lattice (125 words)

	Citebase - Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8 (Site not responding. Last check: 2007-10-22)
		We show that the Leech lattice gives a sphere covering which is locally least dense among lattice coverings.
		The new lattice yields a sphere covering which is more than 12% less dense than the formerly best known given by the lattice A8*.
		Currently, the Leech lattice is the first and only known example of a locally optimal lattice covering having a non-simplicial Delone subdivision.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0405441 (231 words)

	Jerusalem Mathematics Colloquium (Site not responding. Last check: 2007-10-22)
		A strongly perfect lattice is a lattice whose shortest vectors form a spherical 4-design.
		All strongly perfect lattices give local optimal lattice sphere packings and many prominent lattices (the 4-dimensional root lattice D_4, the 8-dimensional root lattice E_8 and the Leech lattice in dimension 24) are strongly perfect.
		Although a precise statement of this principle is not yet established we give some examples: the lattices D_4 and E_8 give local pessima whereas the lattice A_2 and the Leech lattice give local optima.
	www.ma.huji.ac.il /~colloq/2004-05/col.050616.html (292 words)

	LMS Proceedings Abstract, paper PLMS 1498 (Site not responding. Last check: 2007-10-22)
		The alternating group of degree 6 in the geometry of the Leech lattice and K3 surfaces
		The alternating group of degree 6 is located at the junction of three series of simple non-commutative groups: simple sporadic groups, alternating groups and simple groups of Lie type.
		We shall study its new roles both in a finite geometry of a certain pentagon in the Leech lattice and also in the complex algebraic geometry of K3 surfaces.
	www.lms.ac.uk /publications/proceedings/abstracts/p1498a.html (104 words)

	Octonion Products
		The /\24 Leech lattice has 3x240 + 3x16x240 + 3x16x16x240 = = 720 + 11,520 + 184,320 = 196,560 units, related to the 24-dimensional Golay code.
		The 196,560 Leech lattice units, plus 300 = symmetric part of 24x24, plus 24 produce 196,884 which is the dimension of a representation space of the Monster, the largest sporadic finite simple group.
		The Euclidean lattices in 8, 16, and 24 dimensions have Lorentz counterpart lattices in 10, 18, and 26 dimensions.
	www.valdostamuseum.org /hamsmith/480op.html (3069 words)

	School of Mathematics
		In that paper, the heat flow method was used to obtain the rank one case of Lieb's fundamental theorem concerning exhaustion by gaussians; we extend the technique to the higher rank case, giving two new proofs of the general rank case of Lieb's theorem.
		The Leech lattice $\Lambda$ was discovered by John Leech in 1965 in connection with the packing of identical spheres into 24--dimensional space ${\mathbb R}^{24}$ so that their centres lie at the lattice points.
		The Conway group $\cdot O$ is the group of automorphisms of the Leech lattice (fixing the origin).
	www.mat.bham.ac.uk /research/pure/seminars_Autumn05.htm (1091 words)

	CLASSICAL and QUANTUM INFORMATION THEORY
		By going down to the underlying 24-dim Leech lattice space, it should be possible to represent the Monster on the 24-dim space of the Leech lattice.
		A common fundamental structure causes quantum-error-correcting codes to be based on GF(4), the hexacode H6 to be related to the Golay codes and Leech lattice, and an RNA code to be based on 4 nucleotides UGAC, taken 3 at a time.
		Lattices, such as the Leech lattice, are related to tilings.
	www.tony5m17h.net /info.html (6749 words)

	oct124
		I created this octonion calculator to facilitate my continuing casual investigation of the Leech lattice.
		a third octonion such the the trio lies in the 24-d laminated lattice (Leech).
		the Leech lattice (since a = c = 1 when the program opens, bd = a(b(cd)) at the start (see element at right)).
	www.7stones.com /Homepage/Publisher/oct124.html (191 words)

	IFA FA
		The 4^12 = (2^2)^12 structure is analogous to the Complex 12-dimensional structure of the Leech lattice.
		The 16^6 = (2^4)^6 structure is analogous to the Quaternionic 6-dimensional structure of the Leech lattice.
		The 256^3 = (2^8)^3 structure is analogous to the Octonionic 3-dimensional structure of the Leech lattice.
	valdostamuseum.org /hamsmith/VodouFA.html (4863 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Around 1965 John Leech was working on what we now call the Leech Lattice.
		Leech knew that this symmetry group would be interested but he didn’t have the group theory skills needed to do his research.
		This work on the Leech Lattice transformed Conway’s career, it was his first step into mathematical fame.
	www.sienahts.edu /~dm101651/ConwayWork.htm (493 words)

	Theory of Computation at Harvard (Site not responding. Last check: 2007-10-22)
		I will give an overview of the proof that the Leech lattice gives the densest lattice packing in 24 dimensions.
		Furthermore, we show that no sphere packing in that dimension can exceed the Leech lattice's density by a factor of more than $1+1.65 \cdot 10^{-30}$.
		The proof involves various combinatorial properties of the Leech and $E_8$ lattices, as well as computer verification of the properties of certain polynomials.
	www.eecs.harvard.edu /theory/03-04/kumarabs.html (108 words)

	Fields medalist Richard E. Borcherds
		The Leech lattice, which plays an important role in Borcherds’ work as well, is the unique lattice in the 24-dimensional Euclidean space, with a fundamental domain of unit volume, such that the squared length of any element in the lattice is an even integer and there are no lattice elements of squared length 2.
		An example of a vertex operator algebra is given by the Fock space of a string propagating on a torus.
		The moonshine module is obtained by combining a twisted as well as an untwisted vertex operator module associated to the Leech lattice, and amounts to a theory of a string propagating on an orbifold that is not a torus.
	log24.com /log/saved/030528-Borcherds.html (1181 words)

	Lattices: Construction and Operations
		Dual of a lattice, dual quotient of a lattice
		Several interesting lattices are directly accessible inside Magma using standard constructions, e.g., root lattices and preimages of linear codes.
		For each lattice, a LLL-reduced basis for the lattice is computed and stored internally when necessary and subsequently used for many operations.
	magma.maths.usyd.edu.au /magma/Features/node167.html (173 words)

	Re: The Monster and the Leech
		Moreover, the Leech lattice is the only 24-dimensional even self-dual lattice with this property!
		I think the answer must be: yes, the orthogonal symmetry group of the Leech lattice has no elements with det = -1.
		In dimension 26 we get infinitely many fundamental roots forming a Dynkin diagram with one node for each point in the Leech lattice, and edges in a pattern that depends only on the distances between these points in the Leech lattice.
	www.lns.cornell.edu /spr/2000-09/msg0028528.html (615 words)

	leech lattice generator matrix
		Hi there I am reading the paper Amrani, O. et al, "The Leech Lattice and the Golay Code: Bounded-Distance Decoding and Multilevel Constructions", IEEE Transactions on Information Theory, vol.
		It is about leech lattice decoding I wonder what the generation matrix for this kind of leech lattice definition.
		And in SPLAG chap 4, there is a leech lattice generator matrix based on the MOG, I wonder whether it is the generator matrix based on Golay code.
	www.usenet.com /newsgroups/sci.math/msg00047.html (109 words)

	DC MetaData for: Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8 (Site not responding. Last check: 2007-10-22)
		Abstract: We show that the Leech lattice gives a sphere covering which is locally least dense among lattice coverings.
		For this we construct a less dense covering lattice whose Delone subdivision has a common refinement with the Delone subdivision of $\mathsf{E}_8$.
		new lattice yields a sphere covering which is more than $12\%$ less dense than the formerly best known given by the lattice $\mathsf{A}_8^*$.
	www.math.uni-magdeburg.de /preprints/shadows/04-29report.html (209 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		In (finite?) reflection groups, i.e., those that can be generated by a set of their own reflections, the set of all roots contains subsets that form systems of "simple" roots which satisfy certain mutual obtuseness conditions.
		If so, does this mean the orthogonal symmetry group of the Leech lattice has no det=1 operations, i.e., that it's a pure rotation group?
		So..., the Leech's symmetry group, which I believe is called Co_0 (right?) must be either Co_1 x Z_2 or the covering group of Co_1 (right?).
	www.math.niu.edu /~rusin/known-math/00_incoming/leech (372 words)

	The Monster and the Leech
		In article <8qa6ht$7gn$1@newsflash.concordia.ca>, MCKAY john wrote: >Gerard Westendorp asked: >>John Baez wrote: >>> Of these, the strangest and most beautiful is the Leech >>> lattice.
		The Leech >lattice is characterized by having no roots.
		Yes, it's true that of all the 24 self-dual even lattices in 24-dimensional Euclidean space, only the Leech lattice contains no vectors v with v.v = 2.
	www.lns.cornell.edu /spr/2000-09/msg0028290.html (199 words)

	Hexacode-Based Quantization of the Gaussian Source at 1/2 Bit Per Sample (Site not responding. Last check: 2007-10-22)
		A. Vardy and Y. Be'ery, "Maximum likelihood decoding of the leech lattice", IEEE Trans.
		O. Amrani, Y. Be'ery, A. Vardy, F.-W. Sun and H. van Tilborg, "The leech lattice and the golay code: Bounded-distance decoding and multilevel constructions", IEEE Trans.
		O. Amrani and Y. Be'ery, "Efficient bounded-distance decoding of the hexacode, the golay code and the leech lattice", IEEE Trans.
	www.comsoc.org /comm/private/2001/dec/2056_49comm12-ragot.html (316 words)

	From Error-Correcting Codes through Sphere Packings to Simple Groups - Cambridge University Press
		In turn, this highly symmetric lattice, with each point neighbouring exactly 196,560 other points, suggested the possible presence of new simple groups as groups of symmetries.
		From coding to sphere packing; an introduction to sphere packing; the Leech connection; the origin of Leech's first packing in E24; the matrix for Leech's first packing; the Leech lattice; 3.
		From sphere packing to new simple groups; is there an interesting group in Leech's lattice?; the hard sell of a simple group; twelve hours on Saturday, six on Wednesday; the structure of 0; new simple groups; Appendix 1.
	www.cambridge.org /aus/catalogue/catalogue.asp?isbn=0883850370 (379 words)

	Window Boxes and Container Gardening Indoors and Outdoors
		You may know exactly how much to water the plant but if you have a rainy spell it could be the demise of the mini garden that has no drainage system.
		Fertilize well and often, nutrients in a container can leech out.
		Designer Doyle McCullar and landscape designer Cody Schrey help them realize their dream by upgrading the front and adding beautiful French Window Boxes designed and sold by Hooks and Lattice.
	www.hooksandlattice.com /containergar.html (976 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		\\ August 24th, 2004 \\ VARIOUS LATTICES RELATED TO THE LEECH LATTICE (dimensions 2 to 24) \\=========================================================================== \\ CONTENT.
		Antilaminations of L"11mid", orthogonal in Leech of L_{13}^mid \\ 4.
		VARIA (large values of the invariant gamma') \\ All these lattices are contained in the Leech lattice L24 (Lambda_{24}) \\ AFTER each Gram matrix, one can read its standard invariants \\ and the order of its automorphism group \\ NOTE.
	www.math.u-bordeaux.fr /~martinet/Lambda.gp (1045 words)

	The theta functions of sublattices of the Leech lattice, Takeshi Kondo, Takashi Tasaka
		The theta functions of sublattices of the Leech lattice, Takeshi Kondo, Takashi Tasaka
		The theta functions of sublattices of the Leech lattice
		[I] Conway, J. H., A characterization of Leech's lattice, Invent.
	projecteuclid.org /getRecord?id=euclid.nmj/1118780340 (225 words)

	deep holes of leech lattice
		Hello Is there any algorithm to calculate the deep holes of leech lattice?
		Re: deep holes of leech lattice, Robin Chapman
		Re: deep holes of leech lattice, "Stephen M. Fortescue"
	www.usenet.com /newsgroups/sci.math/msg09762.html (92 words)

	Orbits in the Leech Lattice, Daniel Allcock
		We provide an algorithm for determining whether two vectors in the Leech lattice are equivalent under its isometry group, the Conway group $\co0$ of order $\sim8\times10^{18}$.
		We also give algorithms for testing equivalence under these two subgroups.
		We describe our intended applications to the symmetry groups of Lorentzian lattices and the enumeration of lattices of dimension ${}\sim24$ with good properties such as having small determinant.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.em/1136926978 (197 words)

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