On the Shortest Lattice Vector vs. the Shortest Basis

05/31/2023

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by   Yael Eisenberg, et al.

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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 ℓ^q norm, for q∈ℕ_+ (albeit, via another lattice, of a somewhat larger dimension).