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

Abstract: 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.