An Algebraic Decoding Algorithm for Convolutional Codes

The class of convolutional codes generalizes the class of linear block codes in a natural way. The construction of convolutional codes which have a large free distance and which come with an e cient decoding algorithm is a major task. Contrary to the situation of linear block codes there exists only very few algebraic construction of convolutional codes. It is the purpose of this article to introduce a new iterative algebraic decoding algorithm which is capable of decoding convolutional codes which have a certain underlying algebraic structure. The algorithm exploits the algebraic structure of the convolutional code and it achieves its best performance if some naturally associated block codes can be e ciently decoded in an algebraic manner. In order to achieve this goal we will work with a classical state space description of a so called systematic encoder. Using this description we will derive a general procedure which will allow one to extend known decoding algorithms for block codes (like e.g. the Berlekamp Massey algorithm) to convolutional codes. In the coding literature there exist several decoding algorithms for convolutional codes. Maybe the most prominent one is the Viterbi decoding algorithm which applies the principle of dynamic programming to compute the transmitted message sequence. It was shown by Forney [6] that this algorithm computes the message sequence in a maximum likelihood fashion. The disadvantage of this algorithm is that it becomes computationally infeasible if the degree of the encoder is larger than 20. On the side of the Viterbi algorithm there are several sub-optimal algorithms and we would like to mention Massey's threshold decoding algorithm [9], the sequential decoding algorithm and the feedback decoding algorithm [7, 8, 12]. More recently there has been a signi cant interest in some iterative decoding algorithms in connection with the decoding of low density parity check codes and other codes de ned on general graphs and we refer to [17, 20]. The iterative decoding algorithm which we will present in this paper seems to be di erent from above ideas. Indeed the algorithm iteratively computes the state vector xt inside the trellis diagram (see [7, 8]) by making use of the algebraic structure of the convolutional code.