On the left primeness of some polynomial matrices with applications to convolutional codes

Maximum distance profile (MDP) convolutional codes have the property that their column distances are as large as possible for given rate and degree. There exists a well-known criterion to check whether a code is MDP using the generator or the parity-check matrix of the code. In this paper, we show that under the assumption that $n-k$ divides $\delta$ or $k$ divides $\delta$, a polynomial matrix that fulfills the MDP criterion is actually always left prime. In particular, when $k$ divides $\delta$, this implies that each MDP convolutional code is noncatastrophic. Moreover, when $n-k$ and $k$ do not divide $\delta$, we show that the MDP criterion is in general not enough to ensure left primeness. In this case, with one more assumption, we still can guarantee the result.