  Let $\wedge$ be a lattice in $\mathbb{R}^n$ reduced in the sense of Korkine
and Zolotareff having a basis of the form
$(A_1,0,0,\ldots,0),(a_{2,1},A_2,0,\ldots,0)$,
$\ldots,(a_{n,1},a_{n,2},\ldots,a_{n,n-1},A_n)$ where $A_1, A_2,\ldots,A_n$ are
all positive. A well known conjecture of Woods in Geometry of Numbers asserts
that if $A_{1}A_{2}\cdots A_{n}=1$ and $A_{i}\leqslant A_{1}$ for each $i$ then
any closed sphere in $\mathbb{R}^n $ of radius $ \sqrt{n}/2$ contains a point
of $\wedge$. Woods' Conjecture is known to be true for $n\leq 9$. In this paper
we give estimates on the Conjecture of Woods for $10\leq n\leq33$, improving
the earlier best known results of Hans-Gill et al. These lead to an
improvement, for these values of $n$, to the estimates on the long standing
classical conjecture of Minkowski on the product of $n$ non-homogeneous linear
forms.
