Algorithmic List Decoding at Capacity and Optimal Proximity Gaps for Reed-Solomon Codes

RESUMO

We give a unified hidden-derivative framework for list decoding and mutual correlated agreement of ordinary Reed--Solomon codes over prime fields, on arbitrary prescribed evaluation sets. For every fixed slack $\gamma>0$, every sufficiently large block length $n$, every prime $q\ge n$, and every dimension $1\le k\le(1-\gamma)n$, a deterministic algorithm finds all codewords within relative distance $1-k/n-\gamma$ in $q^{O_\gamma(1)}$ time. The final list has size $n^{O_\gamma(1)}$, independently of $q$. Both statements extend to bounded-input-list recovery, with constants depending additionally on the input-list bound. For every fixed curve degree $\ell$, at most $n^{O_{\gamma,\ell}(1)}$ parameters on a curve $f_0+zf_1+\cdots+z^\ell f_\ell$ admit a nearby codeword whose exact agreement support is not a maximal jointly explained support of the coefficient words. For lines this gives MCA error $n^{O_\gamma(1)}/q$, with no proximity loss. The interpolation stage reparameterizes and optimizes the hidden-derivative construction of Brakensiek, Chen, Putterman, Zhang, and Zheng; differential root enumeration uses Kopparty's algorithm. We then prove that a specialization-safe differential equation has a cover by constant-dimensional varieties of polynomial cumulative degree, outside polynomially many parameter values. Intersecting these varieties with equations from the full agreement support yields both the field-size-independent list bound and exact-support MCA.