Best short rate 1/2 tailbiting codes for the bit-error rate criterion

We give a method of finding convolutional codes with minimum bit-error rate (BER) that combines ideas of importance sampling, Monte Carlo integration, and maximum a posteriori probability decoding. The method is applied to rate 1/2 tailbiting convolutional coding, both feedforward and feedback systematic. Tables of BER-minimizing encoders are given for memories 2-5 and tailbiting size 5-40, over a range of good and bad binary symmetric and additive white Gaussian noise channels. The best generators for these cases are in general all different and are not necessarily the generators that optimize distance. The best generators for bad channels are always systematic. The best when the channel quality is known are usually feedforward, but when it is unknown, they are feedback systematic. The best generators in good channels are predicted by a union bound technique.