The smallest open length for rank-metric intersecting codes: does an [8,3]_(q⁶/q) rank-metric intersecting code exist?

Abstract

A rank-metric code C ⊆ 𝔽_(qᵐ)ⁿ is rank-metric intersecting when the rank supports of any two nonzero codewords meet nontrivially (Bartoli, Borello, Marino and Scotti, arXiv:2507.00569); a nondegenerate [n,k,d]_(qᵐ/q) code has this property if and only if its q-system U ⊆ 𝔽_(qᵐ)ᵏ is not 2-spannable, i.e. U ≠ (H₁ ∩ U)+(H₂ ∩ U) for every pair of 𝔽_(qᵐ)-hyperplanes. Borello, Polverino and Zullo (arXiv:2604.02004) settle the admissible lengths for k=3, m=6 except one: such codes exist for 5 ≤ n ≤ 7 and n=9, cannot exist for n<5 or n>9, and the existence of an [8,3]_(q⁶/q) rank-metric intersecting code is left open; they report that an extensive search for [8,3]_(64/2) codes obtained by puncturing their extremal [9,3,5]_(64/2) example fails. We study the cell at q=2 through a finite model of PG(2,64) and a criterion that decides 2-spannability from the largest hyperplane sections alone. We certify the known part of the row (the [9,3,5]_(64/2) example, whose q-system is maximum scattered, and explicit [5,3,3], [6,3,3], [7,3,4] codes over 𝔽₆₄/𝔽₂), turn the reported puncturing search into a theorem — all 511 rank-metric puncturings of the example are 2-spannable, each with an explicit spanning pair of hyperplanes — and prove that no [8,3]_(64/2) rank-metric intersecting code has an 𝔽₄-linear q-system, exhaustively over 1365 systems that cover every equivalence class of such systems. The exclusion itself is routine, as we point out: an 𝔽₄-linear system of 𝔽₂-dimension 8 in 𝔽₆₄³ meets some hyperplane in dimension 6, so its code has d ≤ 2<k; what the exhaustive theorem adds is the certificates and the completeness of the list. A double count yields an identity for the hyperplane weights of every 8-dimensional 𝔽₂-subspace of 𝔽₆₄³, which forces an intersecting [8,3]_(64/2) system to have exactly 31+4n₂+28n₃ hyperplanes of section size 16, pairwise meeting nontrivially; the best candidate produced by a search over 5.5 · 10⁷ subspaces is scattered, has exactly 31 such hyperplanes and still fails on 108 of their 465 pairs, and it is certified as such. The cell itself remains open; non-existence is not claimed. Every finite statement is verified in Lean 4, without Mathlib.

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