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.

中文翻译：

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一些多项式矩阵的左素性在卷积码中的应用

最大距离分布（MDP）卷积码的特性是，对于给定的速率和程度，它们的列距离尽可能大。存在一个众所周知的标准来使用生成器或代码的奇偶校验矩阵来检查代码是否是 MDP。在本文中，我们证明在 $nk$ 除以 $\delta$ 或 $k$ 除以 $\delta$ 的假设下，满足 MDP 准则的多项式矩阵实际上总是左素数。特别是，当 $k$ 除以 $\delta$ 时，这意味着每个 MDP 卷积码都是非灾难性的。此外，当 $nk$ 和 $k$ 不划分 $\delta$ 时，我们表明 MDP 标准通常不足以确保左素数。在这种情况下，再做一个假设，我们仍然可以保证结果。