Edition 2
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Coding Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
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.
Edition 2
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Coding Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
We study binary-input average-radius list recovery using a configuration formed by independent affine squares sharing their origin. If an F_q-linear code in (F_q^s)^n has dimension k, then, for t≥1, c≥0, d=2t+c, and M=1+3t+c, it contains M distinct words and input lists of size at most two with mean relative error at most 1-2/M-min{2n,n+2tu,(4t+c)u}/(Mn), where k≥ d and u=⌊(k-d)/s⌋. Allocating unequal shortening groups can strengthen the equal-group bound; the asymptotic improvement occurs when coloop directions are present. For sixteen words this gives the asymptotic converse 7/8-(5/4)min{R,1/10}, uniformly in the folding parameter and field. We also prove that the associated all-field two-fiber intersection-dimension certificate has exact cost 4t+c. Finally, standard full-length folded Reed–Solomon codes realize the sixteen-word profile with mean error exactly 3/4, including families at the fixed rate 5/48 and arbitrarily large folding and length. These results rule out the average-radius extrapolation of the Boolean-cube curve for output cap fifteen, and analogous extrapolations for larger power-of-two lists. They do not refute the ordinary list-recovery conjecture of Chen and Zhang. The increment is the sparse independent-square profile and its sharp certificate calculation; shortening, independent kernel representatives, and local-profile methods have established predecessors.