Basisradius

Basisradius is a term used to describe a size-related measure of a basis for a vector space or lattice, typically with respect to a chosen norm. In lattice theory, for a lattice L in R^n with a basis B = {b1, ..., bn} under the Euclidean norm, the basis radius Br(B) is defined as the maximum of the norms of the basis vectors: Br(B) = max_i ||bi||. The basis radius of the lattice itself is the minimum possible Br(B) over all bases of L, denoted Br_min(L) = min_B Br(B). This captures how long the vectors in a basis must be to generate the lattice, with a smaller Br_min indicating a shorter, often better-conditioned basis.

Computation and use: Exact minimization of Br_min(L) is generally difficult, so practical work relies on lattice reduction algorithms. The LLL (Lenstra–Lenstra–Lovász) algorithm produces a reduced basis with provable guarantees that bounds the basis radius relative to the lattice’s shortest vector and dimension. Such reduced bases are valuable in algorithms for integer relation detection, cryptanalysis, and other computational tasks involving lattices. The concept can be adapted to other norms beyond Euclidean (Lp norms) or to different centers (balls around a subspace), yielding variant forms of basis radius.

Relation to other concepts: Basisradius is related to, but distinct from, the covering radius and the successive minima. It focuses on the lengths of basis vectors rather than the overall space-filling properties of the lattice. In pure linear algebra, some discussions define Br(B) as the maximum vector length in a given basis, while Br(V) might denote the minimal such value across all bases of a subspace V.

Notes: Basisradius is not universally standardized; in some texts similar ideas appear under “short basis” or “basis reduction.”