Walsh–Hadamard encoding and relative Hamming distance

For $n\ge 0$ and $u\in {\mathbb{F}}_2^n$, the Walsh–Hadamard encoding ${\mathrm{WH}}_n(u)$ is the truth table of the linear function $r\mapsto u\cdot r$ on ${\mathbb{F}}_2^n$, with coordinates indexed in lexicographic order by $r$. Thus the table has length $2^n$, including the one-entry table ${\mathrm{WH}}_0(())=(0)$ when $n=0$. Its entry at a unit vector is ${\mathrm{WH}}_n(u)(e_i)=u\cdot e_i=u_i$.

For two tables $a,b$ with the same nonempty finite coordinate set $I$, their relative Hamming distance is $\mathrm{dist}(a,b)=\frac{\mathrm{\mid }\{i\in I:a(i)\mathrm{\ne }b(i)\}\mathrm{\mid }}{\mathrm{\mid }I\mathrm{\mid }}\mathrm{.}$ For Walsh–Hadamard tables of dimension $n$, $I={\mathbb{F}}_2^n$ and the denominator is $2^n>0$. In particular the $n=0$ table has a well-defined relative distance. The linear functions and their truth-table convention are those of The BLR linearity test over F_2, and the coordinates are the cube variables used by Hadamard linearity constraint system.