Talk:Lenstra–Lenstra–Lovász lattice basis reduction algorithm

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Requested move

Lenstra-Lenstra-Lovász lattice reduction algorithm → Lenstra-Lenstra-Lovász lattice basis reduction algorithm … Rationale: The basis is reduced, not the lattice. … Please share your opinion at Talk:Lenstra-Lenstra-Lovász lattice reduction algorithm. Nuesken 17:24, 18 May 2006 (UTC)[reply]

2nd shortest not equal 2nd successive minimum

I attempted to clarify text related to some issues:

There is not 'the' shortest vector, since there are always at least two of them
One has to be very careful about stuff like "2nd shortest vectors", as such talk is often incorrect for dimension >= 5. For example, consider the parity lattice defined by $\{(x_{i})\mid x_{i}\sim x_{j}\mod 2\forall i,j\}$ . It has $n$ linearly independent vectors of length 1, yet its last successive minimum $\lambda _{n}={\sqrt {n}}>1$ (once the dimension n is >= 5). 131.234.72.252 (talk) 13:52, 16 June 2008 (UTC)[reply]
I apologize, what I wrote yesterday in a hurry is obviously incorrect. The parity lattice has a full set of linearly independent vectors of length 2 (not 1), and there are no shorter nonzero vectors in the lattice (for dimension >= 5). However, those vectors do not form a lattice basis, because the vectors of odd parity cannot be written as integer linear combinations of vectors of even parity. 89.245.87.65 (talk) 06:24, 17 June 2008 (UTC)[reply]

Description of the algorithm

It seems to me that a description of the actual algorithm (at least in high-level pseudocode) is needed in order for this article to be in accord with its title. —Preceding unsigned comment added by 84.97.180.156 (talk) 23:09, 22 November 2009 (UTC)[reply]

Lattice algorithms in general

I agree that more needs to be said about what the algorithm does. Moreover, I fail to see how a lattice, which is a partial ordering on a set of points, can have a basis in the first place.

Grandfather Gauss saw everything mathematical as embedded in some numerical space. As far as I kmow he only ever gave credit to another mathematician once in his lifetime - Bernhard Reimann.

He definitely did not expect the consequences of finite permutation groups (polynomials >=5) as outlined by Galois. But even to this day everyone embeds finite problems in larger spaces.

This simply is not discrete mathematics, and it addresses lattice traversal in a roundabout way. Reply: j.heyden@bigpond.com