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Two codewords $c_1$ and $c_2$ of length $n$, with elements in $\lbrace +1, -1 \rbrace$, and Hamming distance $d$, have a cross-correlation given by $$(n-d) -d = n-2d.$$ The reason is that there are $n-d$ bits that are equal and their product is $1$, and $d$ bits that are different and their product is $-1$. Note that: The larger the distance $d$, the ...
Well, as you can easily verify, these two criteria aren't the same if you define "minimum correlation" to mean that the absolute value of the correlation coefficient is minimized (i.e. 0): In $\mathbb F_2^N$, the vector that's the farthest away from any given vector $v$ is its bit-wise inverse $\overline v$ (using Hamming distance) Using your mapping, the ...