A Distance Measure Tailored to Tailbiting Codes

The error-correcting capability of tailbiting codes generated by convolutional encoders is described. In order to obtain a description beyond what the minimum distance $d_{\min}$ of the tailbiting code implies, the active tailbiting segment distance is introduced. The description of correctable error patterns via active distances leads to an upper bound on the decoding block error probability of tailbiting codes. The necessary length of a tailbiting code so that its minimum distance is equal to the free distance $d_{\mathrm{free}}$ of the convolutional code encoded by the same encoder is easily obtained from the active tailbiting segment distance. This is useful when designing and analyzing concatenated convolutional codes with component codes that are terminated using the tailbiting method. Lower bounds on the active tailbiting segment distance and an upper bound on the ratio between the tailbiting length and memory of the convolutional generator matrix such that $d_{\min}$ equals $d_{\mathrm{free}}$ are derived. Furthermore, affine lower bounds on the active tailbiting segment distance suggest that good tailbiting codes are generated by convolutional encoders with large active-distance slopes.