Some new constructions for simplex codes

Three constructions for n-dimensional regular simplex codes α_i, 0⩽i⩽n, are proposed, two of which have the property that α_i for 1⩽i⩽n is a cyclic shift of α₁. The first method is shown to work for all the positive integers n=1,2,... using only three real values. It turns out that these values are rational whenever n+1 is a square of some integer. Whenever a (v,k,λ) cyclic (or Abelian) difference set exists, this method is generalized so that a similar method is shown to work with ν=n (the number of dimensions)