抄録
A finite subset X of the n-dimensional simplex is called a simplex t-design if the integral of any polynomial of degree at most t over the simplex is equal to the average value of the polynomial over the set X. Although these designs on a simplex are tightly connected to several other topics in mathematics, such as spherical designs, an explicit construction of such designs is not well-studied. In this paper, we will explicitly construct such designs using a union of sets consisting of elements whose coordinates are a cyclic permutation of a particular point. By choosing such a set, the conditions of a set to be a simplex t-design can be reduced to a system of t equations. Solving these system of equations, we managed to explicitly construct simplex 2-designs on a simplex of an arbitrary dimension.
