Defect of an octahedron in a rational lattice

Consider an arbitrary $n$-dimensional lattice $\mathrm{\Lambda }$ such that ${\mathbb{Z}}^n\subset \mathrm{\Lambda }\subset {\mathbb{Q}}^n$. Such lattices are called {\it rational} and can always be obtained by adding $m\le n$ rational vectors to ${\mathbb{Z}}^n$. {\it Defect } $d(\mathcal{E},\mathrm{\Lambda })$ of the standard basis $\mathcal{E}$ 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 $\mathcal{E}$ together with some $d$ vectors from the lattice $\mathrm{\Lambda }$ form a basis of $\mathrm{\Lambda }$. Let $\mathrm{\mid }\mathrm{\mid }\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{\mid }\mathrm{\mid }$ be $L^1$-norm on ${\mathbb{Q}}^n$. Suppose that for each non-integer $x\in \mathrm{\Lambda }$ inequality $\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }>1$ holds. Then the unit octahedron $O^n=\{x\in {\mathbb{R}}^n:\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }⩽1\}$ is called admissible with respect to $\mathrm{\Lambda }$ and $d(\mathcal{E},\mathrm{\Lambda })$ is also called defect of the octahedron $O^n$ with respect to $\mathcal{E}$ and is denoted as $d(O_{\mathcal{E}}^n,\mathrm{\Lambda })$. Let $d_n^m={\max }_{\mathrm{\Lambda }\in {\mathcal{A}}_m}d(O_{\mathcal{E}}^n,\mathrm{\Lambda }),$ where ${\mathcal{A}}_m$ 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 $m<n$ $$d_n^m\le C\frac{n\ln (m+1)}{\ln \frac{n}{m}}{(\ln \ln {\left(\frac{n}{m}\right)}^m)}^2$$ 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.