Abstract
A geometric method of lattice reduction based on cycles of directional and hyperplanar shears is presented. The deviation from cubicity at each step of the reduction is evaluated by a parameter called `basis rhombicity' which is the sum of the absolute values of the elements of the metric tensor associated with the basis. The levels of reduction are quite similar to those obtained with the Lenstra-Lenstra-Lovász (LLL) algorithm, at least up to the moderate dimensions that have been tested (lower than 20). The method can be used to reduce unit cells attached to given hyperplanes.
Highlights
In a recent paper (Cayron, 2021), we proposed a method to determine a unit cell attached to any hyperplane p
The unit cell attached to the hyperplane p is made of one short vector b1 pointing to a node of the first layer parallel to the plane p, i.e. such that the scalar product ptb1 1⁄4 1, and of N À 1 short vectors fb2; . . . ; bi; . . . ; bNg lying in the plane p, i.e. such that the scalar product ptbi 1⁄4 0, where ‘t’ means ‘transpose’
Even if the vectors fb1; . . . ; bi; . . . ; bNg determined by the algorithm are already quite short, they can be reduced even more, i.e. it is possible to find shorter vectors fb01; . . . ; b0i; . . . ; b0Ng defining a smaller and more orthogonal unit cell of the same volume associated with the same hyperplane p, i.e. fulfilling the same Bezout’s identity and integer relation
Summary
In a recent paper (Cayron, 2021), we proposed a method to determine a unit cell attached to any hyperplane p. The unit cell attached to the hyperplane p is made of one short vector b1 pointing to a node of the first layer parallel to the plane p, i.e. such that the scalar product ptb1 1⁄4 1, and of N À 1 short vectors fb2; . The first vector is a solution of Bezout’s identity, and the N À 1 vectors are solutions of the integer relation, both with the coordinates pi. ; bNg determined by the algorithm are already quite short, they can be reduced even more, i.e. it is possible to find shorter vectors fb01; . ; b0Ng defining a smaller and more orthogonal unit cell of the same volume associated with the same hyperplane p, i.e. fulfilling the same Bezout’s identity and integer relation. Reducing the length of the vectors in a lattice is related to the general problem called ‘lattice reduction’
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