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We propose a transmission protocol and the corresponding coding scheme for a cognitive bidirectional cooperative communication system consisting of two primary nodes, two secondary nodes, and a half-duplex relay node. The two primary nodes, as well as the two secondary nodes, transmit to each other with the relay assistance, but no information exchange is needed between a primary node and a secondary node. The proposed protocol for the system under consideration is based on a six-stage coding scheme for the single-pair bidirectional relay system, also proposed in this paper, which exhibits a larger achievable rate region than an existing four-stage coding scheme. The proposed protocol provides a larger achievable sum rate region for the primary and secondary pairs, compared with the scheme where the primary and secondary pairs perform the relay-assisted bidirectional transmission in a time-division manner. Moreover, we design Raptor codes for the system under consideration to approach the limits given by the achievable rate region. We decompose the entire code graph into various code blocks and design the code profiles for each block. Numerical evaluations and simulation results demonstrate the performance improvement of the proposed protocol, and the capacity achieving performance of the optimized Raptor codes.