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In this correspondence, it is shown that some convolutional codes with distinct memory sizes of minimal encoders are equivalent in the sense that the minimal trellises of these codes are the shifted versions of one another. For an (n, k) binary convolutional code, the weight spectrum obtained from the minimal trellis may be slightly different from that obtained from the conventional code trellis with n-bit branches. Code search is conducted to find some good (n, n - 1) binary convolutional codes. Bounds on the trellis complexity, measured by the number of states and the number of branches in the minimal trellis module, of any convolutional code and its equivalent codes are also derived. Index Terms--Error-correction coding, convolutional codes, minimal trellis, minimal trellis module, trellis complexity.

We show that the state complexity profile of a convolutional code C is the same as that of the reciprocal of the dual code of C in case that minimal encoders for both codes are used. Then, we propose an optimum permutation for any given (n, n - 1) binary convolutional code that will yield an equivalent code with the lowest state complexity. With this permutation, we are able to find many (n, n - 1) binary convolutional codes which are better than punctured convolutional codes of the same code rate and memory size by either lower decoding complexity or better weight spectra. Index Terms--Convolutional codes, decoding, trellis codes.