We examine the main linear coding theory problem and study the structure of optimal linear codes over the ring ${mathbb{Z}}_m$. We derive bounds on the maximum Hamming weight of these codes. We give bounds on the best linear codes over ${mathbb{Z}}_8$ and ${mathbb{Z}}_9$ of lengths up to 6. We determine the minimum distances of optimal linear codes over ${mathbb{Z}}_4$ for lengths up to 7. Some examples of optimal codes are given.