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

2026-07-12

AI Summary

This work investigates lower bounds on the cross-consistency error of linear codes in batched proximity testing. By a constructive approach, it establishes—for the first time—a quantitative connection between (p,L)-list decoding counterexamples and cross-consistency errors: such counterexamples are transformed into associated codes that preserve the original algebraic structure. Leveraging tools from finite-field linear algebra, Riemann–Roch spaces, and coordinate-index-preserving transformations, the authors explicitly construct codeword pairs that witness the derived lower bound. For algebraic geometry codes, the cross-consistency error is shown to be at least (1/q)⌈(L+1)N/(N + L·deg G)⌉; a corresponding bound is also obtained for Reed–Solomon codes. These results fit within a general analytical framework that respects the structural properties of the underlying code families.

Abstract

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.

Problem

• list-decoding
• mutual correlated agreement
• lower bounds
• linear codes
• proximity testing

Innovation

• list-decoding counterexamples
• mutual correlated agreement
• structure-preserving construction
• algebraic-geometry codes
• Reed–Solomon codes