Spectral Refutations of Semirandom $k$-LIN over Larger Fields

We study the problem of strongly refuting semirandom $k$-LIN$(\mathbb{F})$ instances: systems of $k$-sparse inhomogeneous linear equations over a finite field $\mathbb{F}$. For the case of $\mathbb{F} = \mathbb{F}_2$, this is the well-studied problem of refuting semirandom instances of $k$-XOR, where the works of [GKM22,HKM23] establish a tight trade-off between runtime and clause density for refutation: for any choice of a parameter $\ell$, they give an $n^{O(\ell)}$-time algorithm to certify that there is no assignment that can satisfy more than $\frac{1}{2} + \varepsilon$-fraction of constraints in a semirandom $k$-XOR instance, provided that the instance has $O(n) \cdot \left(\frac{n}{\ell}\right)^{k/2 - 1} \log n /\varepsilon^4$ constraints, and the work of [KMOW17] provides good evidence that this tight up to a $\mathrm{polylog}(n)$ factor via lower bounds for the Sum-of-Squares hierarchy. However for larger fields, the only known results for this problem are established via black-box reductions to the case of $\mathbb{F}_2$, resulting in an $|{\mathbb{F}}|^{3k}$ gap between the current best upper and lower bounds. In this paper, we give an algorithm for refuting semirandom $k$-LIN$(\mathbb{F})$ instances with the "correct" dependence on the field size $|{\mathbb{F}}|$. For any choice of a parameter $\ell$, our algorithm runs in $(|{\mathbb{F}}|n)^{O(\ell)}$-time and strongly refutes semirandom $k$-LIN$(\mathbb{F})$ instances with at least $O(n) \cdot \left(\frac{|{\mathbb{F}^*}| n}{\ell}\right)^{k/2 - 1} \log(n |{\mathbb{F}^*}|) /\varepsilon^4$ constraints. We give good evidence that this dependence on the field size $|{\mathbb{F}}|$ is optimal by proving a lower bound for the Sum-of-Squares hierarchy that matches this threshold up to a $\mathrm{polylog}(n |{\mathbb{F}^*}|)$ factor. Our results also extend to the more general case of finite Abelian groups.