Eight-point fiber certificates and binary-input list recovery of folded Reed–Solomon codes
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Abstract
For eight distinct points over an arbitrary field, we construct a finite nonnegative rational certificate bounding the total population of the two largest fibers of every linear quotient. The certificate has intercept two and total dimension cost at most nine, and its weights are fixed independently of the quotient. This cost is attained by the Boolean three-cube. A standard folded-Wronskian argument then gives average-radius list recovery with input lists of size two and output cap seven for primitive-power folded Reed–Solomon codes whenever (k-1)/(n(s-6))<2/3-8ρ/9, with folding s≥7. The asymptotic tradeoff is the corresponding case of the existing Chen–Zhang conjecture. As supporting results, we prove that three arbitrary replacements of vertices of a Boolean four-cube preserve certificate cost at most 56/3, and exhibit an all-field family of affine dimension six attaining that cost.
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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
- No additional permission granted
Scope
All-field eight-point quotient-fiber certificates with intercept two and dimension cost at most nine, and the resulting binary-input/output-cap-seven average-radius bound for the stated primitive-power folded Reed–Solomon regime, with structured sixteen-point extensions.
Disclosures
OpenAI assistants supported argument development, literature discovery, proof checking and drafting. The named author is responsible for the mathematical arguments and the manuscript.
Reproduction
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