The tailbiting BCJR decoder executes the forward and backward recursions around and around a tailbiting convolutional code trellis. We will motivate why tailbiting codes and the TB-BCJR are important in practice, and summarize results in the last 5 years. The TB-BCJR decoder must converge to a certain pair of eigenvectors; however, it is not a true MAP decoder. We discuss when its convergence is very close and compare the TB-BCJR to related decoders including the true MAP. Other topics include best encoders for different E_b/N₀ and circle sizes, and practical means to achieve the forward/backward convergence.

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