The chromatic number of the Grassmann graph J₂(6,3) lies between 19 and 24, and no published representative of a known APN class on 𝔽₂⁶ is third-order sum-free

Abstract

A function F:𝔽₂ⁿ → 𝔽₂ᵐ is kth-order sum-free if the sum of its values over every affine k-dimensional flat of 𝔽₂ⁿ is nonzero; for k=2 this is almost perfect nonlinearity. Heering, Kaspers and Taranchuk showed that a {k-1,k}-order sum-free (n,m)-function of algebraic degree k yields a proper colouring of the Grassmann graph J₂(n+1,k) with 2ᵐ-1 colours, and obtained from the cubic function x⁷ on 𝔽_(2⁵) the bound χ(J₂(6,3)) ≤ 31, the best known. We give a colouring of J₂(6,3) with 24 colours: an explicit partition of the 1395 planes of PG(5,2) into 24 constant-dimension subspace codes of minimum subspace distance 4, of sizes 51 to 76, all within the maximum A₂(6,4;3)=77 of Honold, Kiermaier and Kurz. Spending that same 77 in the ratio bound gives χ(J₂(6,3)) ≥ 19, so the published window 15 ≤ χ(J₂(6,3)) ≤ 31 becomes 19 ≤ χ(J₂(6,3)) ≤ 24; the exact value remains open, and the 24-colouring is a certificate found by search, not a construction, so it says nothing about J₂(n,3) for other n. On the source paper's own open problem — do APN and third-order sum-free functions on 𝔽_(2ⁿ) exist for n>5? — we verify their n=5 examples and settle the published 6-bit data: the six APN power maps, the six Kim-type maps, and the published representative of each of the 14 known CCZ-classes on 𝔽_(2⁶) are all APN and none is third-order sum-free. This is not a nonexistence statement, and we say exactly why. Along the way the base-64 representative table we decoded is found to appear misaligned with its own invariant columns by five rows. Every theorem below is machine-checked in Lean 4.

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