There exist Singleton-like upper bounds for the column distances of convolutional codes, which can be reached if the underlying finite field is sufficiently large. However, it is not known which field size is necessary to reach these bounds and known code constructions require very large field sizes. In the first part of this talk, we inverse this construction problem by fixing the field and investigating what column distances are possible over the given field. In this way we construct binary convolutional codes with maximal column distances among all binary convolutional codes with the same code parameters. Furthermore, as these optimal binary convolutional codes turn out to be related to binary first-order Reed-Muller block codes, we can use this to develop an efficient decoding algorithm for these convolutional codes. This algorithm is a reduced complexity version of the Viterbi algorithm and hence it is a maximum likelihood algorithm. Finally, we generalize the presented construction of optimal binary convolutional codes to arbitrary finite fields, i.e. for some code rates and fixed arbitrary finite field, we obtain a construction of convolutional codes with optimal column distances.
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