Eight-point fiber certificates and binary-input list recovery of folded Reed–Solomon codes
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Abstract
We prove that folded Reed–Solomon codes with primitive-power evaluation blocks and two allowed symbols at each coordinate have output list size at most seven at every asymptotic rate R<2/3-8ρ/9, where 0≤ρ<3/4 is the disagreement fraction. The result holds in the stronger average-radius sense, with explicit finite-parameter folding and field-size conditions. The proof assigns one nonnegative subspace certificate to each configuration of eight messages, controlling the two largest fibers of every linear quotient, including partial captures. Every eight-point configuration over any field has certificate cost at most nine, and the Boolean three-cube attains nine. Combining this inequality with classical folded-Wronskian rank bounds gives the input-two, output-seven case of the Chen–Zhang tradeoff. Retaining partial-overlap windows in the folded determinant gives a quadratic intersection budget and sharper integer finite-folding profiles. These certify, for example, n=10, k=120 and ρ=1/4 at folding 29 over F₅₁₂. The same certificate applies to multiplicity codes in characteristic at least k; its derivative-Wronskian ramification budget certifies folding 28 over F₃₃₁ for the same n,k,ρ. Binary input refers to the two allowed symbols, independently of the field characteristic.
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Research details
- Subjects
- Mathematics · Coding Theory
- Keywords
- folded Reed–Solomon codes, list recovery, affine configurations, fiber certificates, folded Wronskian
- Manuscript date
- 2026-10-18
- Reader reuse
- none
Scope
Average-radius input-two/output-seven list recovery for the specified primitive-power folded Reed–Solomon codes at R<2/3−8ρ/9, via an all-field robust eight-point certificate of cost at most nine. Cyclic partial-window determinant bounds and integer support profiles sharpen finite folding; multiplicity-code ramification applies in characteristic p≥k.
Disclosures
Generative AI tools, including ChatGPT and Codex, assisted research exploration, literature discovery, proof development, checking and manuscript preparation under the author's direction. The named author is responsible for the mathematical content and attribution. The research was self-funded with no external support. The author owns Luminary Oddities and its Research Archive.
Reproduction
Twelve-page manuscript and sixteen-member source package, with eight complete TeX files, portable build helper, exact finite-receiver verifier and expected JSON, reader instructions, canonical citations and checksums. The packaged source reproduces the PDF byte for byte. The verifier reproduces the folded-29 and multiplicity-28 examples and the failed folded-28 control; the general statements are proved analytically.