A general procedure is developed to implement the xed delay tree search FDTS detector in the multi-dimensional signal space. The procedure systematically nds a required set of hyperplanes that divide the entire space into di erent decision regions. A Boolean mapping function is also analytically determined which generates a nal binary decision from several intermediate binary variables that indicate the position of the observation vector relative to given hyperplanes. The proposed technique is demonstrated on a 1 + D 2 partial response channel driven by the 1,k-coded input.