as rapidly as possible for as long as possible are called maximum distance profile (MDP) codes. These codes were introduced in [@strongly] and are especially suitable for the use in sequential decoding algorithms. In [@vp], the authors showed that MDP convolutional codes can correct the maximum possible number of errors in some sliding window of a certain length (depending on the code parameters). Moreover, they considered reverse MDP convolutional codes, which have the advantage that optimal error correction is possible with forward and backward decoding algorithms. Finally, complete MDP convolutional codes, which are again a subclass of reverse MDP convolutional codes, have the additional benefit that they can correct even more error patterns than reverse MDP convolution codes, e.g. there is less waiting time when a large burst of erasures occurs and no correction is possible for some time [@vp].

The existence (and genericity) of reverse MDP convolutional codes for all code parameters has been proven in [@vp]. In [@cmdp], it has been shown that for the existence of an $(n,k,\delta)$ complete MDP convolutional code, it is necessary to have $(n-k)\mid\delta$ and that complete MDP convolutional codes exist (and are generic) for all code parameters fulfilling this condition. The case 