Is the multi-bit gold code resulted from a xor of two different multi-bit m-sequences will still maintain its statistical property (bounded correlation, distribution of 1 and 0) ?

  • $\begingroup$ I'm adding that these sequences need to not be identical :) $\endgroup$ Commented Nov 17, 2020 at 10:42

1 Answer 1


Yes, that is the whole point of Gold sequences. If $x$ and $y$ are a pair of preferred m-sequences of period $N$, then their periodic cross-correlation function is bounded. It also happens to be take on only three distinct values. The $N+2$ Gold sequences are

$$x,~~ y,~~ x\oplus y, ~~x\oplus Ty, ~~x\oplus T^2y,~~ \cdots, ~~x \oplus T^{N-1}y$$

(where operator $T$ means periodically or cyclically shift the sequence left by one place) and the periodic cross-correlation function of any pair of (distinct) Gold sequences also takes on only the same three values as the periodic cross-correlation function of $x$ and $y$ (and hence satisfies the same bound). Ditto the periodic autocorrelation function of any Gold sequence (except for the autocorrelation peak $N$ at $0$). However, the distribution of $0$'s and $1$'s in the Gold sequences is not always nearly balanced as in the m-sequences. For some choices of $T^i$, the Gold sequence is nearly balanced, while for other choices, there is a preponderance of $0$'s or a preponderance of $1$'s. See the paper "Cross-correlation properties of pseudorandom and related sequences," Proc. IEEE, vol.68, pp.593-619, May 1980 which I referred you to just yesterday.


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