Abstract. As an attempt to understandmotives over k[x]/(x[supm]), we define the cubical additive higher Chow groups with modulus for all dimensions extending the works of S. Bloch, H. Esnault and K. Rülling on 0-dimensional cycles. We give an explicit construction of regulator maps on the groups of I-cycles with an aid of the residue theory of A. Parshin and V. Lomadze.