列表译码反例给出互相关协议错误的下界
List-Decoding Counterexamples Yield Lower Bounds on Mutual Correlated Agreement Error
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中文总结 AI 辅助
研究表明列表译码反例可给出互相关协议错误的下界。通过给定线性码列表译码性的反例构造相关码，给出错误下界及见证字，还有保持结构版本。并应用于AG评估码和里德 - 所罗门码，得出相应结论。
中文摘要 AI 辅助
互相关协议捕获接收字的随机线性组合是否能与一个码产生新的大一致性，这与批量接近度测试的健全性相关。我们通过构造证明列表译码反例给出了互相关协议错误的下界。给定一个关于\(\mathbb{F}_q\)上线性码的\((p,L)\)-列表译码性的显式反例，我们构造一个相同长度和维度的相关码\(C'\)，使得\(\operatorname{err}_{\mathrm{MCA}}(C',p)\geq\frac{1}{q}\left\lceil\frac{(L + 1)q}{q + L}\right\rceil\)，同时其最小距离最多减少一个。该构造还产生见证此错误的一对显式字。我们进一步为坐标由有限集\(\Omega\)索引的码族给出一个保持结构的版本，每个索引通过映射\(v:\Omega\to\mathbb{F}_q^k\)确定一个生成矩阵列。该构造最多改变一个坐标索引，并确保输出码保持在相同的索引族中。作为应用，我们将此原理应用于代数几何（AG）评估码和里德 - 所罗门码。对于AG码，如果\(G\)是定义基础黎曼 - 罗赫空间的除子，\(N\)是可用于评估的\(\operatorname{supp}(G)\)之外的有理点数量，所得码保持在相同的函数域和黎曼 - 罗赫空间中，具有修改后的评估位置集。其互相关协议错误至少为\(\frac{1}{q}\left\lceil\frac{(L + 1)N}{N + L\mathrm{deg} G}\right\rceil\)。里德 - 所罗门码的结论通过范德蒙德列特化得出。
英文摘要
Mutual correlated agreement captures whether a random linear combination of received words can create a new large agreement with a code, a property relevant to the soundness of batched proximity testing. We show constructively that list-decoding counterexamples yield lower bounds on the mutual correlated agreement error. Given an explicit counterexample to the $(p,L)$-list-decodability of a linear code over $\mathbb{F}_q$, we construct a related code $C'$ of the same length and dimension such that $\operatorname{err}_{\mathrm{MCA}}(C',p)\ge\frac{1}{q}\left\lceil\frac{(L+1)q}{q+L}\right\rceil$, while decreasing its minimum distance by at most one. The construction also produces an explicit pair of words witnessing this error. We further give a structure-preserving version for code families whose coordinates are indexed by a finite set $Ω$, with each index determining a generator-matrix column through a map $v:Ω\to\mathbb{F}_q^k$. The construction changes at most one coordinate index and ensures that the output code remains in the same indexed family. As applications, we instantiate this principle for algebraic-geometry (AG) evaluation codes and Reed--Solomon codes. For AG codes, if $G$ is the divisor defining the underlying Riemann--Roch space and $N$ is the number of rational places outside $\operatorname{supp}(G)$ available for evaluation, the resulting code remains over the same function field and Riemann--Roch space, with a modified set of evaluation places. Its mutual correlated agreement error is at least $\frac{1}{q}\left\lceil\frac{(L+1)N}{N+L\mathrm{deg} G}\right\rceil$. The Reed--Solomon conclusion follows as the Vandermonde-column specialization.
