In the quantum network, a biphoton source can be utilized to transmit Hadamard codes with high coincidence rates. It is the strong correlation in the biphoton state that facilitates the encoding and decoding task. We propose to multiplex the biphoton state from the cascade emissions of cold atomic ensembles under the four-wave mixing condition, and use this effective biphoton state to fulfill the encoding and decoding task of Hadamard codes. Using this highly correlated biphoton source with frequency entanglement, we obtain the coincidence rates and quantify the performance by calculating their contrasts. In this biphoton state, we explore various ways to encode and decode the weight and phase degrees of freedom in multiplexing to identify the best performance. In addition, non-orthogonal codes are introduced to manifest a better performance in the task of encoding and decoding. We find that these codes can enlarge the number of code lengths and promise a high-capacity quantum communication of Hadamard codes, useful in quantum error correction application.
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