, Volume 10, Issue 4, pp 333-348

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

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

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.