# Multi-bit gold code

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) ?

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

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.