Rank-Two Frobenius-Linearized Normal Forms and Orthoderivative Dual Coordinates in Quadratic APN Maps

Abstract

We classify binary-linear two-term Frobenius-linearized operators $L(Y)=AY^σ+BY$ on $K^3$, where $K$ is a finite extension of $\mathbb{F}_2$ and $σ$ 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 $σ$, to the canonical model $(α,β,γ)\mapsto(α^σ+α,β^σ,γ)$. 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 $σ$-quadratic almost perfect nonlinear maps, this identifies the orthoderivative by $π_F(X)^T F(X)=1$; 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.