A simplified criterion for MDP convolutional codes
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Maximum distance profile (MDP) convolutional codes have the property that their column distances are as large as possible. There exists a well-known criterion to check whether a code is MDP using the parity-check matrix of the code. However, this criterion has only been shown under the assumption that the parity-check matrix is left prime. Proving left primeness of a polynomial matrix is not an easy task and also, in almost all the previous papers which provide a construction of MDP convolutional codes, there is a gap in the application of the mentioned criterion to show the MDP property, since it is not proven that the parity-check matrix is left prime. In this paper we close this gap. In particular, we show that under the assumption that (n-k)|δ or k|δ, a polynomial matrix that fulfills the MDP criterion is actually always left prime. Moreover, when (n-k)∤δ and k∤δ, we show that the MDP criterion is not enough to ensure left primeness. In this case, with one more assumption, we can guarantee the result.
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