All Research

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.