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This paper investigates tail-biting trellis realizations for linear blockcodes. Intrinsic trellis properties are used to characterize irreducibility ongiven intervals of the time axis. It proves beneficial to always consider thetrellis and its dual simultaneously. A major role is played by trellisproperties that amount to observability and controllability for fragments ofthe trellis of various lengths. For fragments of length less than the minimumspan length of the code it is shown that fragment observability and fragmentcontrollability are equivalent to irreducibility. For reducible trellises, aconstructive reduction procedure is presented. The considerations also lead toa characterization for when the dual of a trellis allows a productfactorization into elementary ("atomic") trellises.

Local Irreducibility of Tail-Biting Trellises