$(n-k)\nmid\delta$, in which a complete MDP convolutional code cannot exist, is also much more involved when considering just MDP convolutional codes, see [@dr]. There are some general constructions for MDP [@strongly], [@dr13] and complete MDP [@cmdp] convolutional codes. However, all of these constructions have the disadvantage that they only work over base fields of very large size.

This provokes the question for the minimal field size such that an MDP respective complete MDP convolutional code could exist. For the case of MDP convolutional codes, there is something done to solve this problem in [@b], where the authors provide an upper bound on the necessary field size. In [@b] as well as in [@strongly], where an - until now unproven - conjecture about a bound on the necessary field size is raised, superregular Toeplitz matrices are used, i.e. the question is connected to the problem of determining the necessary field size for the existence of such superregular Toeplitz matrices. In this paper, we improve these bounds by other means than using superregular Toeplitz matrices. For complete MDP convolutional codes, the so far only result on the necessary field size could be derived from the constructions in [@cmdp] but this is leading 