Reed-Solomon 码在容量下的算法列表译码与最优接近间隙
Algorithmic List Decoding at Capacity and Optimal Proximity Gaps for Reed-Solomon Codes
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- University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
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中文总结 AI 辅助
本文提出统一隐藏导数框架，实现 Reed-Solomon 码在容量下的确定性列表译码，并证明最优接近间隙，同时给出与域大小无关的列表界和精确支持 MCA 结果。
中文摘要 AI 辅助
我们针对素数域上的普通 Reed-Solomon 码，在任意指定的求值集上，给出了用于列表译码和相互一致协议（MCA）的统一隐藏导数框架。对于每个固定的松弛参数 $\gamma>0$、每个足够大的块长 $n$、每个素数 $q\ge n$ 以及每个维度 $1\le k\le(1-\gamma)n$，一个确定性算法在 $q^{O_\gamma(1)}$ 时间内找到相对距离 $1-k/n-\gamma$ 内的所有码字。最终列表的大小为 $n^{O_\gamma(1)}$，与 $q$ 无关。这两个结论都扩展到有界输入列表恢复，其常数额外依赖于输入列表的界。对于每个固定的曲线次数 $\ell$，在曲线 $f_0+zf_1+\cdots+z^\ell f_\ell$ 上至多有 $n^{O_{\gamma,\ell}(1)}$ 个参数允许存在一个邻近码字，其精确一致支持集不是系数字的极大联合解释支持集。对于直线，这给出了 MCA 误差 $n^{O_\gamma(1)}/q$，且没有接近损失。插值阶段重新参数化并优化了 Brakensiek、Chen、Putterman、Zhang 和 Zheng 的隐藏导数构造；微分根枚举使用 Kopparty 的算法。然后我们证明，一个专门安全的微分方程在多项式个参数值之外，被具有多项式累积次数的常数维簇所覆盖。将这些簇与来自完全一致支持集的方程相交，得到与域大小无关的列表界和精确支持 MCA。
英文摘要
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 $γ>0$, every sufficiently large block length $n$, every prime $q\ge n$, and every dimension $1\le k\le(1-γ)n$, a deterministic algorithm finds all codewords within relative distance $1-k/n-γ$ in $q^{O_γ(1)}$ time. The final list has size $n^{O_γ(1)}$, 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 $n^{O_{γ,\ell}(1)}$ 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 $n^{O_γ(1)}/q$, 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.
