As cyclic codes and maximum distance separable (MDS) codes, cyclic MDS codes have very nice structures and properties, which have been intensively investigated in literature due to their theoretical interest and practical importance. Particularly, abundant cyclic MDS codes have been determined and constructed for many parameters and most of them were proved to be equivalent to generalized Reed-Solomon (GRS) codes. Hence it is a challenging task to construct non-Reed-Solomon type cyclic MDS codes. In this work, we obtain many new cyclic MDS codes for certain parameters by determining the solutions of the system of polynomial equations. Moreover, by determining the dimension of the Schur square of an MDS code, we can easily show that all of our constructed codes are not equivalent to GRS codes.

One of the main problems of subspace coding is to determine the maximal size of a constant dimension subspace code with given parameters. In this paper, we show that mixed dimension subspace codes can be used to construct large constant dimension subspace codes. We introduce a new class of subspace codes called mixed dimension/distance subspace codes. Using such codes, we present two constructions for large constant dimension subspace codes. The problem about the sizes of our constant dimension subspace codes is transformed into finding mixed dimension/distance subspace codes with large dimension distributions. The new constructed codes are the largest known for many sets of parameters. Our method gives at least 136 new lower bounds on the sizes of constant dimension subspace codes.

The main problem of constant-dimension subspace coding is to construct constant-dimension codes (CDCs) with the maximum possible cardinality. Lifting Ferrers diagram codes is an effective way to construct CDCs. In addition, the discovery of linkage construction allows to construct many CDCs, i.e., lower bounds. In this paper, we combine the two methods of construction and obtain some new lower bounds of CDCs for small parameters.