Tales of Tailbiting Codes

Abstract: Tailbiting codes are error correcting codes and can be used in digital transmission systems. They are obtained by terminating convolutional codes into block codes. This thesis consists of an introduction to convolutional and tailbiting codes, and the Papers A--G that include the main results. In Paper A, the error correcting capability of tailbiting codes is described via the active distances for convolutional codes. This leads to a new upper bound on the block error probability of tailbiting codes. A lower bound on the minimum distance as a function of the length of a tailbiting code is given. It can be applied when designing concatenated coding schemes. An efficient algorithm that computes coefficients of the distance spectrum of a convolutional code or a tailbiting code is presented in Paper B. It was used in code searches resulting in tables of convolutional and tailbiting codes that are superior to the best previously known codes. Furthermore, two new maximum-likelihood decoding algorithms for tailbiting codes are presented. Tailbiting codes are in general generated by convolutional encoders with large active distance-slopes. In Paper C, bounds on the active distance-slope are presented. Moreover, the active distance-slope is used as a search criterion to find tailbiting codes that are better than previously known codes. In Paper D, the active distances are used to analyze the error correcting capability of convolutional codes when the encoder starting and ending states are unknown to the decoder. This description provides insight into suboptimal decoding algorithms for tailbiting codes. It is demonstrated in Paper E that the quasi-cyclic behavior of tailbiting codes can be exploited to decrease their error probabilities when decoded suboptimally. Moreover, it is shown that a large active distance-slope has a positive effect on the performance of suboptimal tailbiting decoders. Woven convolutional codes (WCC) are concatenated codes with many interesting properties. In Paper F, encoder and distance properties of WCCs with tailbiting component codes are investigated. Finally, in Paper G, error exponents are derived for these WCCs.