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While in most literature the decoding complexity of convolutional codes has been measured by the encoder memory size, in this paper we study convolutional codes under the total number of edge symbols per information bit in the minimal trellis module representing them as the measure of decoding complexity. We conduct a code search restricted to the recently introduced class of generalized punctured convolutional codes (GPCC), which is broad enough to contain good codes and yet has structural properties that facilitate the code search. For the same decoding complexity and the same code rate, new codes are compared to well-known existing classes of convolutional codes such as punctured convolutional codes (PCCs) and unit-memory codes (UMCs). The comparison is extended to convolutional codes recently found by Rosnes and Ytrehus (RY), who have considered the maximum entry of the state complexity profile of the minimal trellis module as the trellis complexity in their code search. As reflected by the comparison carried out in this paper, the best (in a distance spectrum sense) convolutional codes of existing and new trellis complexities are tabulated. We note that, for the same decoding complexity and the same code rate, some of the new codes have larger free Hamming distance than the existing UMCs, while some other new codes have slightly improved distance spectrum as compared to best PCCs. On the other hand, some of the RY codes are better than the best GPCCs searched up to the present time.