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This paper classifies the scalar trace-product switchings of the Gold function \(x \mapsto x^3\) in even dimensions, revealing that nontrivial switchings exist only for dimensions 4, 6, and 8, with specific admissible coefficients identified for each. The authors provide a uniform proof across dimensions and establish a dimension-rigidity theorem, demonstrating that no nonzero coefficients are admissible for even dimensions greater than 10. Additionally, they classify normalized rank-two extensions in dimension eight and describe the structure of the associated projective classes, significantly advancing the understanding of projective geometry in this context.
Nontrivial trace-product switchings of the Gold function are confined to just three even dimensions, challenging assumptions about their prevalence in higher dimensions.
We completely classify a natural scalar trace-product switching of the Gold almost perfect nonlinear function $x\mapsto x^3$ in every even dimension. Nontrivial switchings occur only for $n=4,6,8$: the admissible coefficients are, respectively, the nonzero trace-zero elements, the six elements of multiplicative order nine, and $\mathbb{F}_4^{*}$. For every even $n\geq10$, no nonzero coefficient is admissible. The infinite range is excluded by additive-character estimates on a Fermat cubic, with exact finite bridges for $n=10,12$. The raw coefficient lists for $n=6,8$ appeared earlier in Arshad's dissertation; our contribution is their intrinsic description, a proof uniform in the dimension, and the resulting dimension-rigidity theorem. We also classify normalized rank-two extensions in dimension eight by $\mathbb{P}^{1}(\mathbb{F}_4)$. A binary trace selector accepts two coefficient values at each non-base projective point, and the eight accepted marked switchings form exactly two extended-affine, hence two CCZ, classes. A centre-independent low-rank derivative criterion reduces each rank-$r$ candidate to $2^r-1$ membership tests in precomputed forbidden sets. The global APN classes reached are known; the results describe their local organization around the Gold centre and rule out this switching mechanism in all larger even dimensions.
