The trellis complexity of convolutional codes

It has long been known that convolutional codes have a natural, regular trellis structure that facilitates the implementation of Viterbi's algorithm [30,10]. It has gradually become apparent that linear block codes also/]ave a natural, though not in general a regular, "minimal" trellis structure, which allows them to be decoded with a Viterbi-1ike algorithn] [2,31,22,11,27,14,12,16,24,25,8,15]. In both cases, the complexity of tile Viterbi decoding algorithm can be accurately estimated by the number of trellis edges per encoded bit. It would, therefore, appear that we are in a good position to make a fair comparison of the Viterbi decoding complexity of block and convolutional codes. Unfortunately, however, this comparison is somewhat muddled by the fact that some convolutional codes, tile punctured convolutional codes [4], are known to have trellis representations that are sig_Jificantly less complex than the conventional trellis. In other words, ttle conventional trel]is representation for a convolutional code may not be the minimal trellis representation. Thus, ironically, at present we seem to know more about the minimal trel]is representation for block than for convolutional codes. In this article, we provide a remedy, by developing a theory of minimal trellises for convolutional codes. (A similar theory has recently been given by Sidorenko and Zyabloy [29].) This allows us to make a direct performal_ce-complexity comparison for block and convolutional codes. A by-product of our work is an algorithm for choosing, from among a11 generator matrices for a given convolutional code, what we call a trellis-minimal generator matrix, from which the minimal trellis for the code can be directly constructed. Another by-product is that, in the new theory, punctured convolutional codes no longer appear as a special class, but simply as high-rate convolutional codes whose trellis complexity is unexpectedly small.