  Consider an arbitrary $n$-dimensional lattice $\Lambda$ such that
$\mathbb{Z}^n \subset \Lambda \subset \mathbb{Q}^n$. Such lattices are called
{\it rational} and can always be obtained by adding $m \le n$ rational vectors
to $\mathbb{Z}^n$. {\it Defect } $d({\cal E},\Lambda)$ of the standard basis $
{\cal E}$ of ${\mathbb Z}^n$ ($n$ unit vectors going in the directions of the
coordinate axes) is defined as the smallest integer $d$ such that certain $
(n-d) $ vectors from $ {\cal E} $ together with some $d$ vectors from the
lattice $\Lambda$ form a basis of $\Lambda$.
  Let $||...||$ be $L^1$-norm on $\mathbb{Q}^n$. Suppose that for each
non-integer $x \in \Lambda$ inequality $||x|| > 1$ holds. Then the unit
octahedron $O^n = \left\{{ x} \in \mathbb{R}^n: ||x|| \leqslant 1\right\}$ is
called admissible with respect to $\Lambda$ and $d({\cal E},\Lambda)$ is also
called defect of the octahedron $O^n$ with respect to $\cal{E}$ and is denoted
as $d(O^n_{{\cal E}}, \Lambda)$.
  Let $ d_n^m = \max_{\Lambda \in {\cal A}_m} d(O^n_{{\cal E}},\Lambda), $
where $ {\cal A}_m $ is the set of all {\it rational} lattices that can be
obtained by adding $m$ rational vectors to $\mathbb{Z}^n$: $ \Lambda = \left
\langle {\mathbb Z}^n, { a}_1, \dots, { a}_m \right \rangle_{{\mathbb Z}}, {
a}_1, \dots, { a}_m \in {\mathbb Q}^n. $ In this article we show that there
exists an absolute positive constant $ C $ such that for any $m < n $ $$ d_n^m
\leq C \frac{n \ln (m+1)}{\ln \frac{n}{m}} \left(\ln\ln
\left(\frac{n}{m}\right)^m \right)^2 $$
  This bound was also claimed in $[1],[2]$, however the proof was incorrect. In
this article along with giving correct proof we highlight substantial
inaccuracies in those articles.
