List-Decoding Counterexamples Yield Lower Bounds on Mutual Correlated Agreement Error

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 $C^{\mathrm{\prime }}$ of the same length and dimension such that ${\mathrm{err}}_{\mathrm{M}\mathrm{C}\mathrm{A}}(C^{\mathrm{\prime }},p)\ge \frac{1}{q}\lceil \frac{(L+1)q}{q+L}\rceil$, 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 $\mathrm{\Omega }$, with each index determining a generator-matrix column through a map $v:\mathrm{\Omega }\to {\mathbb{F}}_q^k$. 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 $\mathrm{supp}(G)$ 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 $\frac{1}{q}\lceil \frac{(L+1)N}{N+L\mathrm{d}\mathrm{e}\mathrm{g}G}\rceil$. The Reed–Solomon conclusion follows as the Vandermonde-column specialization.