The increasing interest in spatial encoding with non-linear magnetic fields has intensified the need for coils that generates such fields. Matrix coils consisting of multiple coil elements appear to offer a high flexibility in generating customized encoding fields and are particularly promising for localized high resolution imaging applications. However, coil elements of existing matrix coils were primarily designed and constructed for better shimming and therefore are not expected to achieve an optimal performance for local spatial encoding. Moreover, eddy current properties of such coil elements were not fully explored. In this work, an optimization problem is formulated based on the requirement of local non-linear encoding and eddy current reduction that results in novel designs of coil elements for an actively-shielded matrix gradient coil. Two metrics are proposed to assess the performance of different coil element designs. The results are analyzed to reveal new insights into coil element design.