DNA computing is being applied to solve problems in combinatorial optimization, logic and Boolean circuits. Breakthrough solutions in combinatorial optimization are the most impressive area of success but, in order to solve combinatorial optimization problems, problems related to the reliability of biological operators, stable DNA expressions, processing speed, expandability and the universality of evaluation criteria must be solved. This study implements a DNA sequence generation system that minimizes errors using DNA coding based on evolutionary models and performs simulation using biological experiment operators. The usefulness of this system is evaluated by applying the Hamiltonian Path Problem (HPP) in the form of a genetic algorithm. The proposed system generates sequences with minimal errors, as compared to existing systems, and identifies optimal solutions for combinatorial optimization problems in significantly reduced processing times.