In this paper we propose construction of optimum space-time convolutional codes over groups and fields for quasi-static flat fading channels. These codes are constructed from cyclic codes in order to generate the trellis codes. The space-time convolutional codes over groups are constructed from cyclic group codes based on the solution of the quadratic forms associated with the lattices (double-struck Z)2 and (double-struck A)2. The space-time convolutional codes over fields are found from the construction of maximum distance separable Reed-Solomon and BCH codes whose generator polynomials do not contain any binary factor. 133 Conway, J.H., Sloane, N.J.A., (1988) Sphere Packings, Lattices and Groups, , Springer-Verlag, New YorkDe Almeida, C., Palazzo R., Jr., Efficient two-dimensional interleaving technique by use of the set partitioning concept (1996) Electronics Letters, 32 (6), pp. 538-540. , MarchTrumpis, B.D., (1975) Convolutional Coding for M-ary Channels, , Ph.D. Dissertation, System Science Dept., UCLATarokh, V., Seshadri, N., Calderbank, A.R., Space-time codes for high data rate wireless communication: Performance criterion and code construction (1998) IEEE Trans. Inform. Theory, 44 (2), pp. 744-765. , March