Given an arbitrary basis for a mathematical lattice, to find a ``good" basis for it is one of the classic and important algorithmic problems. In this note, we give a new and simpler proof of a theorem by Regavim (arXiv:2106.03183): we construct a 18-dimensional lattice that does not have a basis that satisfies the following two properties simultaneously: 1. The basis includes the shortest non-zero lattice vector. 2. The basis is shortest, that is, minimizes the longest basis vector (alternatively: the sum or the sum-of-squares of the basis vectors). The vectors' length can be measured in any $\ell^q$ norm, for $q\in \mathbb{N}_+$ (albeit, via another lattice, of a somewhat larger dimension).

翻译：给定数学格的一个任意基，寻找其“好”基是经典且重要的算法问题之一。本文给出了Regavim（arXiv:2106.03183）定理的一个全新且更简洁的证明：我们构造了一个18维格，该格不存在同时满足以下两个性质的基：1. 基包含最短非零格向量；2. 基是最短的，即最小化最长基向量（或基向量的和、平方和）。向量的长度可采用任意$\ell^q$范数度量（其中$q\in \mathbb{N}_+$），但需通过另一个维度稍大的格实现。