Paper 2026/891

Interleaving Stability for Mutual Correlated Agreement and Curve Decodability

Abstract

We prove that row-wise interleaving does not impose a linear loss on two coding-theoretic soundness properties used in recent IOP/SNARK analyses: generator mutual correlated agreement and curve decodability. For generator-MCA, let $G:\Omega\to\mathbb{F}_q^\ell$ be a coefficient generator over a finite seed set and let $C$ be an $\mathbb{F}_q$-additive code. For every interleaving width $s$ and distance parameter $\delta$, we show $$ \varepsilon_G(C,\delta) \le \varepsilon_G(C^{\equiv s},\delta) \le \left(1+\frac1q+\cdots+\frac1{q^{s-1}}\right)\varepsilon_G(C,\delta). $$ Moreover, if $|\Omega|\le q$, then the transfer is exact: $$ \varepsilon_G(C^{\equiv s},\delta)=\varepsilon_G(C,\delta). $$ In particular, affine-line MCA is invariant under row-wise interleaving. This answers the known interleaving-loss question and removes the linear interleaving factor from the affine-line MCA bound. It also implies that polynomial-generator MCA bounds transfer to interleaved codes without an additional interleaving-width factor. We further establish interleaving stability for curve decodability. We introduce a marked formulation, prove its equivalence to the standard definition for $\mathbb{F}_q$-additive codes and $1\le b\le a\le q$, and use it to transfer curve decodability to row-wise interleavings. If $C$ is $(\ell,\delta,a,b)$-curve-decodable and $\binom{a}{b}\le q$, then $C^{\equiv s}$ is also $(\ell,\delta,a,b)$-curve-decodable for every $s$. We also give a field-size-weighted variant that transfers larger base-code witness parameters to smaller interleaved-code witness parameters.

Note: This note addresses the interleaving-loss question for affine-line mutual correlated agreement highlighted in the Open Problems in List Decoding and Correlated Agreement paper, which is related to the Proximity Prize program. It also gives row-wise interleaving transfer theorems for generator-MCA and for curve decodability.