We perform a Monte Carlo simulation of irreversible template copolymerization near a chemically heterogeneous surface with a regular distribution of discrete adsorption sites that selectively adsorb from solution one of the two polymerizing monomers and the corresponding chain segments. In the polymerization model, the chain propagation process is simulated by adding individual monomers to the end of growing macroradical. We focus in this paper on the influence of polymerization rate, adsorption energy, and the distance between adsorption sites on the chain conformation and the primary sequence of the resulting two-letter (AB) copolymers and, specifically, on the coupling between polymerization and adsorption. The conditions for the realization of conformation-dependent copolymerization are formulated. For this regime, we observe the formation of a quasiregular copolymer with two types of alternating sections. One of them contains randomly distributed A and B segments. The second one consists mainly of strongly adsorbed A segments. It is found that the average length of the random sections is proportional to the distance between the nearest neighbor adsorption sites. The average length of the A-rich sections is determined by the “adsorption capacity” of adsorption site. By varying the strength of the effective monomer-substrate interaction and the distribution of adsorption sites on the substrate, the copolymers with different surface-induced primary sequences can be designed and synthesized in a controlled fashion.