few choices of (small) parameters, the new bounds for the field size of this paper are better than all bounds existing up to now.

MDP Convolutional Codes
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In this section, we summarize the basic definitions and properties concerning MDP convolutional codes. One way to define a convolutional code is via polynomial generator matrices.

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A **convolutional code** $\mathfrak{C}$ of **rate** $k/n$ is a free $\mathbb F[z]$-submodule of $\mathbb F[z]^n$ of rank $k$. We refer to it as $(n,k,\delta)$ convolutional code.\
There exists $G(z)\in\mathbb F[z]^{n\times k}$ of full column rank such that $$\mathfrak{C}=\{v(z)\in\mathbb F[z]^n\ |\ v(z)=G(z)m(z)\ \text{for some}\ m(z)\in\mathbb F[z]^k\}.$$ $G(z)$ is called **generator matrix** of the code and is unique up to right multiplication with a unimodular matrix $U(z)\in Gl_k(\mathbb F[z])$.\
The **degree** $\delta$ of $\mathfrak{C}$ is defined as the maximal degree of the $k\times k$-minors of $G(z)$. Let $\delta_1,\hdots, \delta_k$ be the column degrees of $G(z)$. Then, $\delta\leq\delta_1+\cdots+\delta_k$ and if $\delta=\delta_1+\cdots+\delta_k$, $G(z)$ is called a **minimal** generator matrix.

There is a generic subclass of convolutional codes that could not only be described by an image representation via generator matrices but also by a kernel representation via the so-called parity-check matrices, which will be introduced in the following. Therefore, 