for which an $(n,n-1,1)$ reverse or complete MDP convolutional code could exist is $n+1$.

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To get reverse or complete MDP convolutional codes with these parameters, one has the additional condition that all entries of $h_1$ have to be nonzero. Thus, there are $(|\mathbb F|-1)^{n+1}\prod_{i=2}^{n}(|\mathbb F|-i)$ such convolutional codes, which could be constructed with the same technique as mentioned before.

Sufficient field size and probability for MDP convolutional codes with arbitrary parameters
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Sufficient field size
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The goal of this subsection is to estimate what field size one needs such that it is possible to construct an MDP convolutional code with given but arbitrary parameters $n$, $k$ and $\delta$.

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Let $g$ be the polynomial that is formed by the product of all not trivially zero fullsize minors of $\mathcal{H}_L$ and has the entries of the coefficient matrices of $H$ as variables. Then, an $(n,k,\delta)$ MDP convolutional code exists if $|\mathbb F|>\deg(g)$.

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It is sufficient to show the existence of an MDP convolutional code with generic row degrees, i.e. $\nu=\lceil\frac{\delta}{n-k}\rceil$ and consider only matrices $H$ of the form of Remark \[kh\]. This means $H_{\nu}$ and $H_{\nu-1}$ are of the forms $H_{\nu}=\left(\begin{array}{ccc}
I_{\delta-(n-k)(\nu-1)} & 0 & \ast \\ 
0 