We propose a local-search based algorithm to design DNA sequence sets that satisfy several combinatorial constraints about hamming-distance criteria. To deal with the constraints in the local search, we adopt elaborate (and dynamic) neighborhood search frameworks called the Variable Neighborhood Search (VNS) and the Variable Depth Search (VDS). Although our algorithm can deal with many types of hamming distance-based constraints and is easy to extend (e.g., also applicable for other constraints), in computational experiments, we succeeded in generating better sequence sets than the ones generated by exiting methods of more specified constraints.