Article

Combinatorica

, Volume 10, Issue 4, pp 333-348

Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice

  • J. C. LagariasAffiliated withAT&T Bell Laboratories
  • , H. W. LenstraJr.Affiliated withDepartment of Mathematics, University of California
  • , C. P. SchnorrAffiliated withUniversitÄt Frankfurt

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Abstract

Letλ i(L), λi(L*) denote the successive minima of a latticeL and its reciprocal latticeL *, and let [b1,..., b n ] be a basis ofL that is reduced in the sense of Korkin and Zolotarev. We prove that
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and
http://static-content.springer.com/image/art%3A10.1007%2FBF02128669/MediaObjects/493_2005_BF02128669_f2.jpg
, where
http://static-content.springer.com/image/art%3A10.1007%2FBF02128669/MediaObjects/493_2005_BF02128669_f3.jpg
andγ j denotes Hermite's constant. As a consequence the inequalities
http://static-content.springer.com/image/art%3A10.1007%2FBF02128669/MediaObjects/493_2005_BF02128669_f4.jpg
are obtained forn≥7. Given a basisB of a latticeL in ℝ m of rankn andx∃ℝ m , we define polynomial time computable quantitiesλ(B) andΜ(x,B) that are lower bounds for λ1(L) andΜ(x,L), whereΜ(x,L) is the Euclidean distance fromx to the closest vector inL. If in additionB is reciprocal to a Korkin-Zolotarev basis ofL *, then λ1(L)≤γ n * λ(B) and
http://static-content.springer.com/image/art%3A10.1007%2FBF02128669/MediaObjects/493_2005_BF02128669_f5.jpg
.

AMS subject classification (1980)

11 H 06 11 H 50