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

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

and

, where

andγ j denotes Hermite's constant. As a consequence the inequalities

are obtained forn≥7. Given a basisB of a latticeL in ℝm of rankn andx∃ℝm, 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

.

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The research of the second author was supported by NSF contract DMS 87-06176. The research of the third author was performed at the University of California, Berkeley, with support from NSF grant 21823, and at AT&T Bell Laboratories.

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Lagarias, J.C., Lenstra, H.W. & Schnorr, C.P. Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica 10, 333–348 (1990). https://doi.org/10.1007/BF02128669

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AMS subject classification (1980)

  • 11 H 06
  • 11 H 50