On generation and enumeration of orthogonal Chebyshev-Frolov lattices

Abstract

We discuss orthogonal Chebyshev-Frolov lattices, their generating matrices and their use in Frolov cubature formula. We give a detailed account on coordinate-permuted systems that lead to fast computation and enumeration of such lattices. In particular, we explain the recurrences identified in (K. Suzuki and T. Yoshiki, Hiroshima Math. J., 49(1):139–159, 2019) via a plain constructive approach exhibiting a new hierarchical basis of polynomials. Dual Chebyshev-Frolov lattices and their generating matrices are also studied. Lattices enumeration in axis-parallel boxes is discussed.