Talk:Leech lattice

It says:

:"It is the unique lattice with the following list of properties:"

I'm wondering if the above should have said "It is the unique lattice in R²⁴ with the following list of properties:" ? Michael Hardy 22:49, 25 Nov 2003 (UTC)

If you include the first property (generation by the columns of a 24 × 24 matrix), then it is indeed unique, and mentioning R²⁴ is redundant. Perhaps I am missing the point? Michael Larsen 02:46, 26 Nov 2003 (UTC)

I've put a note on Talk:Kissing number problem about a contradiction between these two pages. I hope someone knowledgable can fix whichever is wrong. -- Rory ☺ 23:01, May 31, 2004 (UTC)

The following is not completely clear, at least not to me:

:"The Leech lattice can be explicitly constructed [...] the set of coordinates i such that ai belongs to any fixed residue class (mod 4) is a word in the binary Golay code."

How is a set of coordinates a word in a code? I think it means that the 24-bit words formed by setting bit at index i=1 if and only if the corresponding integers a_i belong to a certain residue class (0,1,2,3) should all be words in the binary Golay code, but please correct me if I'm wrong!

I can not understand either of these descriptions of the connection to the Golay code. NormHardy (talk) 01:56, 29 December 2013 (UTC)

I found the signed comment below on the article itself; it looks like it was intended for talk, so I'm moving it here. —David Eppstein (talk) 03:34, 12 February 2009 (UTC)

:Ernst Witt is mentioned. The discoverer of the automorphism group of this lattice (Conway) visited Witt after this discovery. Conway's views should be recorded. This might be best included under this article.

:John McKay66.130.86.141 (talk) 01:57, 12 February 2009 (UTC)

In fact, in Pierre de la Harpe's notes following Boris Venkov's course on the lattices, I've met a much simpler construction.

One can simply take the "reduction mod 2" map from $\backslash Z^n$ to $F\_2^n=(\backslash Z/2\backslash Z)^n$ (that is a lattice, as the Golay code is linear), and contract it $\backslash sqrt\{2\}$ times. Then — in the Golay code the minimum number of units is 8, so, the preimage norm is at least 8, division by 2 gives 4. So, we have a pair (as code was doubly pair), unimodular (as the code was of dimension of half of that of the space) lattice without roots (as there are no norm 2 vectors).

But I'm not sure if these notes are published (and thus that we can use it as a source); have someone met this construction otherwhere? --Burivykh (talk) 08:54, 15 December 2009 (UTC)

Does the Leech lattice have (some generalization of) a Dynkin diagram? --JWB (talk) 21:28, 15 December 2009 (UTC)

: No, AFAIK, it's exactly the opposite situation: this is the first roots-free integer autodual lattice. So, no roots-no Dynkin diagram. --Burivykh (talk) 14:43, 20 December 2009 (UTC)

::Added a statement to the article - please feel free to correct. --JWB (talk) 05:56, 21 December 2009 (UTC)

::: Thanks, it's OK now.

::: On the other hand, I've [http://en.wikipedia.org/w/index.php?title=Leech_lattice&action=historysubmit&diff=335011876&oldid=332996687 removed] the next phrase, as I couldn't understand it in any way so the statement would be believable... --Burivykh (talk) 00:35, 31 December 2009 (UTC)

The criterion is that the squared length, or norm, is greater than 2. Since the Leech lattice is a Type II (even unimodular) lattice, the squared distance between every pair of points is an even integer. The lowest such number greater than two is four, which equates to the condition that the unsquared length is greater than or equal to 2. The 196560 minimal vectors all have norm 4, or length 2.

The Leech lattice can be characterised as the unique even unimodular lattice in 24 or fewer dimensions with no vectors of norm 2.

Calcyman (talk) 10:14, 2 June 2010 (UTC)

This article http://www.neverendingbooks.org/index.php/who-discovered-the-leech-lattice.html by Lieven le Bruyn( who seems to have access to the appropriate material) and has a translation of the German text seems to disagree with the conclusion here in the Wikipedia article

"The Leech lattice was originally discovered by Ernst Witt in 1940, but he did not publish his discovery; see his collected works (Witt 1998, p. 328) for details."

Le Bruyn states, with discussion of the topic, "However, there is very little evidence to support this claim." Should we delete this claim or not? Any thoughts? —Preceding unsigned comment added by Jamzik (talk • contribs) 21:32, 1 December 2010 (UTC)

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