Spherical designs of harmonic index t

Spherical tt-design is a finite subset on sphere such that, for any polynomial of degree at most tt, the average value of the integral on sphere can be replaced by the average value at the finite subset. It is well-known that an equivalent condition of spherical design is given in terms of harmonic polynomials. In this paper, we define a spherical design of harmonic index tt from the viewpoint of this equivalent condition, and we give its construction and a Fisher type lower bound on the cardinality. Also we investigate whether there is a spherical design of harmonic index attaining the bound.