Exact expressions for SIC sets have been found for Hilbert spaces of all dimensions from d = 2 through d = 53 inclusive, and in some higher dimensions as large as d = 5779 , for 115 values of d in all. Furthermore, using the Heisenberg group covariance on Z d × × Z d , numerical solutions have been found for all integers up through d = 193 , and in some larger dimensions up to d = 2208 .