In this paper, the unequal error protection (UEP) capability of convolutional encoders, in terms of the separation vector, is studied from an algebraic viewpoint. A simple procedure is presented for constructing a generator matrix, which is basic and has the largest separation vector for every convolutional code. Such a generator matrix would be desirable, since the corresponding encoder not only achieves UEP optimality, but also avoids undesired catastrophic error propagation. In addition, canonical generator matrices, which are both basic and reduced, are even more preferable for encoding, since they attain the lowest complexity for Viterbi decoding. However, the direct transformation from a UEP-optimal generator matrix to a canonical generator matrix may come with an unexpected loss of the separation vector. We also propose a specific type of transformation matrix that reduces the external degrees of the generator matrices, from which canonical generator matrices can be constructed that include the mitigated degradation of the separation vector. Finally, beneficial UEP convolutional codes that achieve the maximum free distances for the given code parameters are provided.

We propose a hybrid ARQ (HARQ) scheme for which a rate-1/2 error-correcting code with code length equal to the packet length is used in the HARQ operation, where each packet comprises two subpackets of equal lengths. For the HARQ system with constant packet lengths, the utilization of subpackets enables the flexibility of carrying new message (or data) in the retransmissions. Hence, such a system can gain additional throughput as compared to the system using all-redundancy retransmissions under the conditions of moderate to high signal-to-noise ratios (SNR). In the proposed HARQ scheme, each of the subpackets can be chosen as a data part, or a redundancy part, or a function of two subpackets, where the data part or the redundancy part may belong to different codewords of the rate-1/2 error-correcting code. With a proper design of the function subpacket, we can adaptively control the error-correcting capability used in the HARQ operation. The key idea is that there is possibility of recovering three data subpackets in one retransmission and hence increasing the throughput. The price is that in each retransmission, up to three bits are needed in the feedback channel to indicate which subpacket is successfully recovered.

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.

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.