Paper 2026/1663

Five-Bit Axial Subspace Differential Uniformity: Exact Optima and a DDT-Support Realizability Gap

Abstract

Subspace differential uniformity (SDU) measures the concentration of a differential distribution table (DDT) on affine subspaces. We determine the exact optima of the two axial affine-SDU coordinates in dimension five and establish a strict gap between locally admissible support designs and supports realizable by almost perfect nonlinear (APN) permutations. We first prove that every $16$-subset of $\mathbb{F}_2^5$ meets some affine $3$-flat in at least six points. Equality holds precisely for balanced quadratic indicators of polar rank four, forming a single affine orbit represented by $\operatorname{supp}(\operatorname{Tr}(x^3))$. This gives the relaxed axial optimum $12$. We then classify all $900$ ordered monomial-trace candidates $M_{r,s}(u,v)=\operatorname{Tr}(u^r v^s)$: exactly $100$ attain both relaxed axial optima, yielding $20$ labelled arrays and three product-linear types up to transpose. Despite satisfying regularity and zero-vector-sum constraints, every optimal type violates a global necessary condition for DDT realizability: its two-dimensional character transform contains a negative coefficient where a vectorial Walsh square is required. Finally, the published exhaustive affine classification of five-bit APN permutations, together with direct recomputation of all five class representatives, gives realizable axial minimum $14$, attained simultaneously by the $x^{15}$ class. Thus local incidence and vector-sum constraints permit score $12$, whereas genuine APN-permutation DDT supports require score $14$.