Mathematical proof note · Version 1.0 · October 8, 2026
Rank-Support Lifting and the Nonexistence of [8,3]64/2 Intersecting Codes
National Mobile Communications Research Laboratory and the Frontiers Science Center for Mobile Information Communication and Security, Southeast University, Nanjing 211189, China
Abstract
Rank-support intersections depend on compatibility between codewords together with their individual ranks. We express common support through a section–quotient exact sequence and a canonical compatibility map. For independent selected message restrictions, graph extensions with prescribed quotient maps realize every compatibility matrix uniformly. This gives an exact intersection distribution and a sharp affine-family dimension bound. We then prove that no nondegenerate [8,3]_(64/2) rank-metric intersecting code exists. Trace-dual incidence forces a four-dimensional section; normalization and coordinate shears reduce every prospective code to one of 10,032 twenty-dimensional binary parameter spaces. An exhaustive calculation leaves 12,160 parameters satisfying the individual rank conditions, and each has two rank-four codewords with disjoint supports. The classification, both complete enumerations, and all final obstructions are reproducible from the accompanying programs and data.
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Keywords
rank-metric intersecting codes; rank support; binary finite fields; subspace intersection; length-eight codes.
中文关键词：秩度量交码；秩支撑；二元有限域；子空间交；长度为八的码。
Suggested citation
S. Yang, "Rank-Support Lifting and the Nonexistence of [8,3]64/2 Intersecting Codes," mathematical proof note, version 1.0, Oct. 8, 2026. [Online]. Available: https://github.com/stalin1943/Coding-Theory-Proof-Notes/blob/main/binary-rank-metric-length-eight/proof.pdf