Tailbiting trellis representations of linear block codes with an arbitrary sectionalization of the time axis are studied. The notations of regular and irregular tailbiting codes are introduced and their maximal state complexities are lower-bounded. The asymptotic behavior of the derived bound is investigated. Furthermore, for regular tailbiting codes the product state complexity is lower-bounded. Tables of new tailbiting trellis representations of linear block codes of rates 1/2, 1/3, and 1/4 are presented. Almost all found trellises are optimal in the sense of the new bound on the state complexity and for most codes with nonoptimal trellises there exist time-varying trellises which are optimal. Five of our newly found tailbiting codes are better than the previously known linear codes with the same parameters. Four of them are also superior to any previously known nonlinear code with the same parameters. Also, more than 40 other quasi-cyclic codes have been found that improve the parameter set of previously known quasi-cyclic codes.

Tailbiting codes encoded by feedback convolutional encoders are studied. A condition for when tailbiting will work is given and it is described how the encoder starting state can be obtained for feedback encoders in both controller and observer canonical forms. Finally, results from a search for systematic feedback encoders that encode tailbiting codes with good decoding bit error probabilities are presented.

The result of a search for the world's second type II (doubly-even and self-dual) convolutional code is reported. A rate R=4/8, 16-state, time-invariant, convolutional code with free distance d/sub free/=8 was found to be type II. The initial part of its weight spectrum is better than that of the Golay convolutional code (GCC). Generator matrices and path weight enumerators for some other type II convolutional codes are given. By the "wrap-around" technique tail-biting versions of (32, 18, 8) Type II block codes are constructed.

Tabulations of binary systematic and nonsystematic polynomial convolutional encoders with an optimum distance profile for rate 1/2, 1/3, and 1/4 are given. The reported encoders are found by computer searches that optimize over the weight spectra. The free distances for rate 1/3 and 1/4 are compared with Heller's (1968) and Griesmer's (1960) upper bounds.

The results of an extensive search for short and moderate length polynomial convolutional encoders for time-invariant tail-biting representations of block codes at rates R=1/4, 1/3, 1/2, and 2/3 are reported. The tail-biting representations found are typically as good as the best known block codes.