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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 ...


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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 ...


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