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This paper classifies binary-linear two-term Frobenius-linearized operators on \( K^3 \) under specific conditions related to coefficient rank and binary kernel, revealing that such operators can be reduced to a canonical model via invertible linear transformations. The authors demonstrate that for operators with a one-dimensional kernel, the dual output row corresponds to a unique trace-adjoint normal, which has implications for the structure of quadratic almost perfect nonlinear maps. Additionally, the study connects various algebraic constructions, providing insights into permutation behavior and bijections in projective geometry.
Invertible transformations can simplify complex binary-linear operators to a canonical form, revealing deeper connections in the structure of quadratic APN maps.
We classify binary-linear two-term Frobenius-linearized operators $L(Y)=AY^\sigma+BY$ on $K^3$, where $K$ is a finite extension of $\mathbb{F}_2$ and $\sigma$ is a fixed nontrivial Frobenius automorphism of $K$ with fixed field $\mathbb{F}_2$. Under a coefficient-rank and binary-kernel condition, if $A$ and $B$ both have $K$-rank two and $L$ has a one-dimensional kernel over $\mathbb{F}_2$, then invertible $K$-linear input and output changes reduce $L$, for this fixed $\sigma$, to the canonical model $(\alpha,\beta,\gamma)\mapsto(\alpha^\sigma+\alpha,\beta^\sigma,\gamma)$. The proof constructs the coordinate frames from the two coefficient-kernel directions and the binary kernel. In these coordinates, the first dual output row is exactly the unique nonzero trace-adjoint normal, with an exact $K$-valued normalization. For pure $\sigma$-quadratic almost perfect nonlinear maps, this identifies the orthoderivative by $\pi_F(X)^T F(X)=1$; in odd extension degree it also yields permutation behavior and a bijection from the projective plane to its dual. The triprojective construction of Gologlu and Kolsch and the cubic norm-twist construction of Li, Zhou, Li, and Qu provide two realizations arising from different algebraic constructions. The triprojective case further admits a determinant factorization and a complete dual frame, whereas the norm-twist realization shows that the pure-map consequences do not follow from the operator theorem alone. A natural Gold representation has coefficient-rank pair $(3,3)$, delimiting the rank-two subclass. The normal form also supplies exact extension-field labels for known component-radical and Walsh-support relations.
