Dimension Rigidity and Projective Geometry of Trace-Product Switchings of the Gold Cube

• URL: http://arxiv.org/abs/2608.04261v1
• Date: Tue, 04 Aug 2026 22:36:09 GMT
• Title: Dimension Rigidity and Projective Geometry of Trace-Product Switchings of the Gold Cube
• Abstract summary: We classify a natural scalar trace-product switching of the Gold almost perfect nonlinear function $xmapsto x3$ in every even dimension.<n>For every even $ngeq10$, no nonzero coefficient is admissible.<n>We also classify normalized rank-two extensions in dimension eight by $mathbbP1(mathbbF_4)$.
• Score: 20.271194684947282
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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