  A Walsh zero space (WZ space) for $f:F_{2^n}\rightarrow F_{2^n}$ is an
$n$-dimensional vector subspace of $F_{2^n}\times F_{2^n}$ whose all nonzero
elements are Walsh zeros of $f$. We provide several theoretical and
computer-free constructions of WZ spaces for Gold APN functions
$f(x)=x^{2^i+1}$ on $F_{2^n}$ where $n$ is odd and $\gcd(i,n)=1$. We also
provide several constructions of trivially intersecting pairs of such spaces.
We illustrate applications of our constructions that include constructing APN
permutations that are CCZ equivalent to $f$ but not extended affine equivalent
to $f$ or its compositional inverse.
