Mathematical proof note · Version 1.0 · October 8, 2026
The Universal Length-Eight Gap for Rank-Metric Intersecting Codes
National Mobile Communications Research Laboratory and the Frontiers Science Center for Mobile Information Communication and Security, Southeast University, Nanjing 211189, China
Abstract
We prove that a nondegenerate rank-metric intersecting code with parameters [8,3]_(q^6/q) does not exist for any prime power q ≥ 3. The proof reduces the column system to scalar-fibre configurations, closes repeated ordinary sublines and quadratic fibres, and then analyzes the remaining four-dimensional section through local conics, an octic obstruction, and Hermitian graph equations. Exact finite-field reductions close the small fields q=3,4,5,7, while the point bounds and sextic equations close q ≥ 8. The note records the load-bearing algebraic identities and the coverage interfaces for the finite verification files.
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Keywords
rank-metric intersecting codes; length-eight codes; finite fields; finite geometry; Hermitian geometry.
中文关键词：秩度量交码；长度为八的码；有限域；有限几何；厄米特几何。
Suggested citation
S. Yang, "The Universal Length-Eight Gap for Rank-Metric Intersecting Codes," mathematical proof note, version 1.0, Oct. 8, 2026. [Online]. Available: https://github.com/stalin1943/Coding-Theory-Proof-Notes/blob/main/rank-metric-length-eight-general-q/proof.pdf