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

Let Ai(L), Ai(L*) denote the successive minima of a lattice L and its reciprocal lattice L*, and let [b l , . . . , bn] be a basis of L that is reduced in the sense of Korkin and Zolotarev. We prove that [4/(/+ 3)]),i(L) 2 _< [bi[ 2 < [(i + 3)/4])~i(L) 2 and Ibil2An_i+l(L*) 2 <_ [(i + 3)/4][(n i + 4)/417~ 2, where "y~ =min(Tj : 1 < j _< n} and 7j denotes Hermite's constant. As a consequence the inequalities 1 < Ai(L)An_i+x(L* ) < n2/6 are obtained for n > 7. Given a basis B of a lattice L in R m of rank n and x E R m, we define polynomial time computable quantities A(B) and #(x, B) that are lower bounds for A1 (L) and/~(x, L), where/x(x, L) is the Euclidean distance from x to the closest vector in L. If in addition B is reciprocal to a Korkin-Zolotarev basis of L*, then AI(L) < 3,~A(B) n *2 X and #(x, L) 2 _< ( E i = l "ri )~( , B) 2-