Paper 2026/2617

Mutual Correlated Agreement and List-Decoding: Improvements and Separations

Abstract

At a high level our contribution is threefold: we improve the quantitative MCA error bound for polynomial generators, we prove a refinement of the implication from list-decodability to MCA, and we clarify that in general, these two notions are distinct. We show that, for every finite field $\mathbb{F}$, every nonempty seed set $S\subseteq\mathbb{F}$, every linear code $\mathcal{C} \subseteq \mathbb{F}^{n}$ of relative minimum distance $\delta_{\mathcal{C}}$ and every degree $d \geq 1$, the univariate powers generator $G_{d}(x) := (1,x,\dots,x^{d})$ has mutual correlated agreement for $\mathcal{C}$ at relative proximity parameter $\gamma$ with error $O(d\max\{1,n\gamma\}/|S|)$ below the smaller of the unique decoding radius and the generalized Johnson radius $J_{d+2}(\delta_{\mathcal{C}}) := 1-(1-\delta_{\mathcal{C}})^{1/(d+2)}$, with a one-coordinate margin below the latter when it is smaller. Then, we show that list-decodability at radius $p$ with list size $L$ implies MCA slightly above $1-\sqrt{1-p}$ (the improvement on the radius is dependent on the list size). The previous state-of-the-art result of Gao, Cai, Xu and Kan showed that list-decodability at radius $p<\delta_{\mathcal{C}}$ with list size $L$ implies mutual correlated agreement at every proximity parameter strictly below $1-\sqrt{1-p}$. Finally we construct two families of codes that show obstructions to any improvement of the $1-\sqrt{1-p}$ bound, or to strong converse statements. Informally, a bound of the form $\epsilon_{\mathsf{MCA}}(1-(1-p)^\alpha)\leq Q(n, L)/|\mathbb{F}|$ for some polynomial $Q$ and $\alpha > 1/2$ cannot hold for all linear codes $\mathcal{C}$ under the sole hypothesis that $\mathcal{C}$ is $(p,L)$-list-decodable. Similarly, for any distance $0<\delta<1$ and any $1-\sqrt{1-\delta}<p<\gamma<1$, there exists a family of linear codes with fixed positive rate and relative distance tending to $\delta$ such that $|\mathbb{F}|\epsilon_{\mathsf{MCA}}(\gamma)\leq \mathrm{poly}(n)$ but the list size at radius $p$ is $n^{\omega(1)}$. Both counterexample families exist over fields of any characteristic. Both MCA errors are for the line generator.