Breakthrough Tracker record
Parallel preprints claim the corrected Cheng–Murray deep-hole classification
Fang claims to settle the remaining low-dimensional cases and, with earlier high-dimensional results, prove the corrected Cheng–Murray conjecture for full-length Reed–Solomon codes in every nonexceptional dimension. Chen and Huang independently cover every dimension for fields larger than 19,683, while Fan supplies exact classifications for several interpolation-degree ranges and reproducible finite certificates.
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- Stable ID
math-cheng-murray-reed-solomon-deep-holes-2026- Revision
math-cheng-murray-reed-solomon-deep-holes-2026.v1- Field
- Mathematics and theoretical computer science · Coding theory, finite geometry and combinatorics
- Evidence
- Tier 1 · Peer reviewed: No
- Record state
- Provisional · Parallel provisional proof claims
- Last checked
AI role
Fang reports using GPT-6 to polish the exposition and check some main proofs. Fan reports GPT-5.6 Sol and GPT-6 Astra assistance with exploratory proof development, computation, literature searches and polishing. Chen and Huang disclose no substantive AI role in the inspected manuscript. Each author group retains responsibility.
Record details
- Problem or result
- Corrected Cheng–Murray conjecture on deep holes of full-length Reed–Solomon codes
- Authors
- Weijun Fang; Bocong Chen and Jing Huang; Xiang Fan
- Institutions
- Shandong University; South China University of Technology; Guangzhou University; Sun Yat-sen University
- Result date
- Parallel first-version preprints submitted October 21 and 22, 2026
Why it matters
Deep holes describe received words at maximum distance from a code and connect decoding theory to arcs and maximum-distance-separable extensions in finite geometry. A correct full classification would close a problem posed in 2007 while isolating the known characteristic-two exceptions.
Limits
All three manuscripts are recent and unrefereed. Their scopes overlap without being identical: Fang's full claim relies on earlier high-dimensional classification, Chen and Huang impose a field-size threshold, and Fan proves degree-specific and bounded classifications rather than the whole conjecture. The tracker found no external specialist verdict and does not merge those scopes into stronger evidence than the papers supply.
Sources
- Primary: Fang, The Cheng–Murray Conjecture on Deep Holes of Full-Length Reed–Solomon Codes
- Independent parallel preprint: Chen and Huang, Deep holes of full-length Reed–Solomon codes
- Independent parallel preprint: Fan, Uniform bounds and exact classifications for Reed–Solomon deep holes
Correction and revision history
- 2026-10-25 — Added after primary-source, scope, status, AI-role and limitation review.
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