Talk:Lenstra–Lenstra–Lovász lattice basis reduction algorithm

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)

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)
• 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)