The defects of admissible balls and octahedra in a lattice, and systems of generic representatives

Let ${\mathscr E}=O\,\mathbf e_1,\dots,\mathbf e_n$ be the frame of unit coordinate vectors, let $\Lambda \subset \mathbb R^n$ such that ${\mathbb Z}^n\subset \Lambda$, let ${\mathscr O}_{\mathscr E}^n$ be the unit octahedron, and let ${\mathscr B}_{\mathscr E}^n$ be the unit ball. A set $\Omega \in \{{\mathscr O}_{\mathscr E}^n,{\mathscr B}_{\mathscr E}^n\}$ is said to be admissible in $\Lambda$ if $\Omega \cap \Lambda =\{O,\pm \mathbf e_1,\dots ,\pm \mathbf e_n\}$. The defect $d(\Omega;\Lambda)$, with respect to $\Lambda$, of a set $\Omega$ admissible in $\Lambda$ is the smallest number of vectors to be deleted from ${\mathscr E}$ in order that the remaining system can be complemented to a basis in $\Lambda$. Let $d_n(\Omega)=\max _\Lambda d(\Omega;\Lambda)$ and let $d_n^*(\Omega)=\max _\Lambda ^*d(\Omega;\Lambda)$, where the maximum is taken over all $\Lambda$ in the first case and over all $\Lambda$ such that $\Lambda /{\mathbb Z}^n$ is a cyclic group in the second. It is shown that $d_n^*(\Omega)\gg \frac n{\log n}(\log \log n)^2$ and $d_n(\Omega)\geqslant n-c\frac n{\log n}$, where $c$ is an absolute constant.These results are obtained using methods of geometry and combinatorial analysis.