SOME NEW CONSTRUCTIONS FOR SIMPLEX CODES

Three constructions for n-dimensional regular simplex codes alpha(i), 0 less-than-or-equal-to i less-than-or-equal-to n, are proposed, two of which have the property that alpha(i) for 1 less-than-or-equal-to i less-than-or-equal-to n is a cyclic shift of alpha1. 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, lambda) cyclic (or Abelian) difference set exists, this method is generalized so that a similar method is shown to work with v = n (the number of dimensions).