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We prove that the angle between any two Minkowski-reduced basis vectors is more than \({\pi }/{3};\) if the orthogonal defect of 3-dimension lattice is less than \({2}/{\sqrt{3}},\) the Minkowski-reduced basis of the lattice is \({\pi }/{3}\)-orthogonal; if a weakly \(\theta \)-orthogonal basis for a lattice with \(\theta \geqslant {\pi }/{3}\) has been ordered by the Euclidean norm of the vectors, and the minimum length ratio maximum length is more than \(2\cos \theta ,\) the basis is Minkowski reduced. We improve an algorithm used in JPEG CHEst by changing it from heuristic one to deterministic one, furthermore we add a constraint to reduce the number of unimodular matrix that need to determine. abstract environment.

Relations Between Minkowski-Reduced Basis and $$\theta $$-orthogonal Basis of Lattice