A Note on Systematic Tailbiting Encoders

Tailbiting codes encoded by convolutional encoders are studied. An explanation is given for the fact that, at low signal-to-noise ratios, a systematic feedback encoder results in fewer decoding bit errors than a`nonsystematic feedforward encoder for the same tailbiting code. The analysis is based on a recently introduced code property, namely, the weight density of distance-$d$ codewords. For a given distance-$d$ weight density, the decoding bit error probability depends on an encoder property, viz., the number of taps in the tap-minimal encoder pseudoinverse. Among all convolutional encoders that encode a given tailbiting code, the systematic one has the tap-minimal encoder pseudoinverse with fewest taps and, hence, gives the smallest bit error probability.