Lattice Reduction - Nearly Orthogonal

One measure of nearly orthogonal is the orthogonality defect. This compares the product of the lengths of the basis vectors with the volume of the parallelepiped they define. For perfectly orthogonal basis vectors, these quantities would be the same.

Any particular basis of vectors may be represented by a matrix, whose columns are the basis vectors . In the fully dimensional case where the number of basis vectors is equal to the dimension of the space they occupy, this matrix is square, and the volume of the fundamental parallelepiped is simply the absolute value of the determinant of this matrix . If the number of vectors is less than the dimension of the underlying space, then volume is . For a given lattice, this volume is the same (up to sign) for any basis, and hence is referred to as the determinant of the lattice or lattice constant .

The orthogonality defect is the product of the basis vector lengths divided by the parallelepiped volume;

From the geometric definition it may be appreciated that with equality if and only if the basis is orthogonal.

If the lattice reduction problem is defined as finding the basis with the smallest possible defect, then the problem is NP complete. However, there exist polynomial time algorithms to find a basis with defect where c is some constant depending only on the number of basis vectors and the dimension of the underlying space (if different). This is a good enough solution in many practical applications.