convolutional code $\mathfrak{C}$ of rate $k/n$ and degree $\delta$ has **maximum distance profile (MDP)** if $$d_j^c(\mathfrak{C})=(n-k)(j+1)+1\quad \text{for}\ j=0,\hdots,L:=\left\lfloor\frac{\delta}{k}\right\rfloor+\left\lfloor\frac{\delta}{n-k}\right\rfloor$$

As mentioned in the introduction, MDP convolutional codes have the property that their column distances increase as rapidly as possible for as long as possible. Indeed, $j=L$ is the largest possible value for which $d_j^c$ can attain the upper bound from Theorem \[ub\]. Moreover, according to [@strongly], it is sufficient to have equality for $j=L$ in Theorem \[ub\] to get an MDP convolutional code.

In the following, we will provide criteria to check whether a convolutional code has a maximum distance profile. Therefore, we need the notion of trivially zero determinants.

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(i) Let $n\in\mathbb N$ and $A\in\mathbb F^{n\times n}$ be a matrix with the property that each of its entries is either fixed to zero or is a free variable from $\mathbb F$. Its determinant $\det(A)$ is called **trivially zero** if it is zero for all choices for the free variables in $A$.\
(ii) An $n\times n$ Toeplitz matrix of the form $\left(\begin{array}{ccc}
a_1 &  & 0 \\ 
\vdots & \ddots &  \\ 
a_n & \hdots & a_1
\end{array} \right)$ is called **superregular** if all its minors that are not trivially 