we need the notion of right prime and left prime polynomial matrices.

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Let $\overline{\mathbb F}$ denote the algebraic closure of $\mathbb F$. A polynomial matrix $G(z)\in\mathbb F[z]^{n\times k}$ with $k<n$ is called **right prime** if it has full column rank for all $z\in\overline{\mathbb F}$. For $k>n$, it is called **left prime** if it has full row rank for all $z\in\overline{\mathbb F}$.

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A convolutional code $\mathfrak{C}$ is called **non-catastrophic** if one and therefore, each of its generator matrices is right prime.

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If $\mathfrak{C}$ is non-catastrophic, there exists a so-called **parity-check matrix** $H(z) \in\mathbb F[z]^{(n-k)\times n}$ of full rank, such that $$\mathfrak{C} =\{ v(z)\in \mathbb F[z]^n\ |\ H(z)v(z) = 0 \in \mathbb F[z]^{n-k}\}.$$ Clearly, a parity-check matrix of $\mathfrak{C}$ is not unique and it is possible to choose it left prime and row proper. In this case, the sum of the row degrees of $H(z)$ is equal to the degree $\delta$ of $\mathfrak{C}$ [@con].\
$H(z)$ has **generic row degrees** if $\nu=\lceil\frac{\delta}{n-k}\rceil$ and the first $\delta-(n-k)(\nu-1)$ row degrees of $H(z)$ are equal to $\nu$ and the remaining $(n-k)\nu-\delta$ row degrees are equal to $\nu-1$.

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Allowing permutation of the entries of the codeword $v(z)$ respective of the columns of 