As in the preceding subsection, we finally consider the probability for MDP convolutional codes under the condition that the code is non-catastrophic.

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The probability that a non-catastrophic $(2,1,1)$ convolutional code is MDP or reverse MDP is $((1-t^n)(1-t^{n-1}))^{-1}(1-t)^{n+1}\prod_{i=2}^{n-1}(1-it)$.

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That $\begin{pmatrix} g_{0,i}\\ g_{1,i}\end{pmatrix}$ is linear independent to $\begin{pmatrix} g_{0,j}\\ g_{1,j}\end{pmatrix}$ for $j\in\{1,\hdots,i-1\}$ implies that the entries of $G(z)$ are coprime and hence each $(n,1,1)$ MDP convolutional code is non-catastrophic. Thus, to get the conditional probability, one just has to divide the formula of the preceding theorem by the probability of non-catastrophicity, which is $1-t^{n-1}$; see [@ich].

$(n,n-1,1)$ convolutional codes for $n\geq 3$
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As the $(n,n-1,1)$ MDP convolutional codes are dual to the $(n,1,1)$ MDP convolutional codes treated in the previous subsection, we easily get all $(n,n-1,1)$ MDP convolutional codes and know that they exist if and only if $|\mathbb F|\geq n$. For the construction, we just replace the conditions on the generator matrix $G$ from the preceding subsection by the same conditions on the parity-check matrix $H$. The following theorem considers $(n,n-1,1)$ reverse and complete MDP convolutional codes.

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The number of $(n,n-1,1)$ reverse and complete MDP convolutional codes is both $(|\mathbb F|-1)^{n+1}\prod_{i=2}^{n}(|\mathbb F|-i)$. Hence, the minimal field size 