On the Successive Minima of Subbases of Low-Dimensional Lattices

Abstract

In this paper we discuss the global optimality of vector lengths for lattice bases. By introducing a partial order on lattice bases and the concept of successive minimal basis (SMB for short), we show that any of its minimal elements is a general greedy-reduced basis, and its least element (if exists) is an SMB. Furthermore, we prove the existence of SMB for lattices of dimension up to 6.