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Gold code is known to have a good cross-corrleation property, i.e., a high cross-correlation value shows iff two sequences are the same.

I wonder whether it is independent to modulation, e.g., BPSK, QPSK, M-PSK, M-QAM, when Gold code is applied to wireless communication.

My naive guess that, since mapping from a binary Gold code to a modulated signal is 1-to-1, so the answer is yes, independent to modulation.

P.S. I'm a bit confused whether this question is proper to EE.stackexchange or math.stackexchange.

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You are correct that the relative increase in immunity to cross-correlation is the same for any modulation scheme. That is to say, the error rate improvement as a proportion to the error rate from another coding scheme should be the same. You only have to remember that each of the modulation schemes you mention each have their own inherent symbol cross-correlation at the binary level. So the improvement will be there, but each scheme will have it own error rate, related to the length of the Gold code symbol combined with the inherent error rate of the modulation scheme.

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