Let $\pi_i$as the projection to the orthogonal complement of $\left\{b_j\right\}_{j=1}^{i-1}$.Then the basis is **HKZ-reduced** if it is size-reduced and $||b_i^*||=\lambda_1\left(\pi_i(L)\right)$. This definition gives us a relatively simple way to compute a HKZ-reduced basis by iteratively finding the shortest vector in orthogonal projections.

$\frac4{i+3}\leq\left(\frac{||b_i||}{\lambda_i(L)}\right)^2\leq\frac{i+3}4$