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

Letλi(L), λi(L*) denote the successive minima of a latticeL and its reciprocal latticeL*, and let [b1,..., bn] be a basis ofL that is reduced in the sense of Korkin and Zolotarev. We prove that and, where andγj 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)≤γn*λ(B) and.

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