In this paper, we present a method to construct girth-6 protographs that lead to the shortest constraint length in the convolutional structure. A stringent structural constraint is imposed on the protographs such that the decoder can be implemented efficiently. Then, given the structural constraint, it is shown that finding the aforementioned protographs is equivalent to solving a simple algebraic problem. Based on this mathematical formulation, girth-6 protographs are created without having to resort to a graph search. Using the girth-6 protographs, we derive good low-density parity-check (LDPC) convolutional codes by using periodic quasi-cyclic lifting. The performance of such constructed codes is compared with AR4JA-based LDPC convolutional codes.
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