A Note on Performance of Generalized Tail Biting Trellis Codes

Tail biting trellis codes and block concatenated codes are discussed from random coding arguments. Error exponents and decoding complexity for generalized tail biting (GTB) random trellis codes, and their relationships are derived, where the GTB trellis codes consist of full tail biting (FTB) trellis codes, partial tail biting (PTB) trellis codes and direct truncated (DT) trellis codes. We show that the PTB trellis codes at all rates except for low rates are superior among the GTB trellis codes, in a sense that they have smaller upper bound on the probability of decoding error for given decoding complexity. We then propose the generalized version of the block concatenated codes constructed by the GTB trellis inner codes, and derive error exponents and the decoding complexity for the proposed code. The results obtained show that the DT trellis inner codes are effective among the GTB trellis inner codes for constructing the generalized version of the concatenated codes to keep the same decoding complexity as the original concatenated codes. We also show that larger error exponents are obtained by the generalized version of concatenated codes, if the decoding complexity is allowed to be larger than that of the original concatenated code, although it is still in polynomial order.