DNA codes have garnered significant interest due to their utilization in digital media storage, cryptography, and DNA computing. In this paper, we first extend the results of constructing reversible group codes \cite{Cengellenmis} and reversible composite group codes \cite{Korban5} to general even-order finite groups. By using these results, we give parallel searching algorithms to find some new DNA codes with better parameters. Secondly, by mapping codes over $\mathbb{F}_4$ to DNA codes, we establish a relationship between the $GC$-weight enumerator of DNA codes and the Hamming weight enumerator of their trace codes, which greatly improves the computational efficiency of searching for DNA codes. Based on this relationship, we propose an efficient algorithm for generating DNA codes with $50\%$ $GC$-content. Furthermore, we find that there is no direct connection between the $GC$-weight enumerator of a DNA code and the $GC$-weight enumerator of its dual code. Finally, we present algorithms for determining whether a DNA code is free from secondary structures or conflict-free, and some new DNA codes with better parameters under multiple constraints are obtained, which are listed in Tables 1 and 4.