In many applications, e.g., broadcast systems and multimedia communication systems, the source data are of dierent error sensitivities. To make the best use of the channel band- width, it is desirable to design a channel code with the capability of unequal error protection (UEP). Convolutional codes is one of the channel codes which have the UEP capability and more studies on convolutional codes for UEP are developed recently. Except for the UEP capability, coding complexity is also a key element which is dominated by the McMillan degree of a generator matrix for convolutional codes. To minimize the complexity while maintain the optimal UEP capability, in this thesis, we propose a procedure for obtaining a UEP-optimal polynomial generator matrix (PGM) which has the smallest McMillan degree among all UEP-optimal PGMs. However, it is unknown whether the McMillan degree of the resulting PGM is the same as the lowest McMillan degree of all possible UEP-optimal generator matrices. Hence, two sucient conditions for a UEP-optimal PGM to have the smallest McMillan degree among all UEP-optimal generator matrices are derived. We also demonstrate the existence of the above desirable generator matrix for a sub class of convo- lutional codes.