In this paper we prove lower bounds for sizes of refutations of unsatisfiable vector Subset Sum instances $\overrightarrow{a}_1 x_1 + \dots + \overrightarrow{a}_n x_n = \overrightarrow{b}$ in the proof system Res(lin$_{\mathbb{F}_q}$) where $char(\mathbb{F}_{q})\geq 5$. As a basis for the hardness criterion for such instances we choose the property of the matrix $A$ with columns $(\overrightarrow{a}_1, \ldots, \overrightarrow{a}_n)$ to be (the transpose of) the generating matrix for a good error-correcting code $C_{A} := \{x\cdot A\, |\, x \in \mathbb{F}_{q}^k\}\subset \mathbb{F}_{q}^n$ and prove the following lower bounds: 1) For a dag-like fragment of Res(lin$_{\mathbb{F}_q}$). We introduce the notion of $(s,r)$-robustness for Subset Sum instances, which in particular implies that $A$ defines an error-correcting code with the minimal distance $s\geq r$. For $(s,r)$-robust instances we prove $2^{Ω(r)}$ lower bound for sizes of refutations in a dag-like fragment of Res(lin$_{\mathbb{F}_q}$). We show that random instances are $(n / 3, Ω\left((n/(q + 1)\ln q))^{1/3}\right))$-robust and that specific examples achieving these bounds can be constructed using algebraic geometry codes. 2) For tree-like Res(lin$_{\mathbb{F}_q}$) refutations we show the size lower bound $2^{Ω({((q+1)\ln q)^{-1/3}}d^{1/5})}$ for any Subset Sum instance where $d$ is the minimal distance of $C_{A}$.

翻译：本文证明了在证明系统Res(lin$_{\mathbb{F}_q}$)（其中$\operatorname{char}(\mathbb{F}_{q})\geq 5$）中，不可满足向量子集和实例$\overrightarrow{a}_1 x_1 + \dots + \overrightarrow{a}_n x_n = \overrightarrow{b}$的驳斥规模的下界。我们选取矩阵$A$（其列向量为$(\overrightarrow{a}_1, \ldots, \overrightarrow{a}_n)$）作为优良纠错码$C_{A} := \{x\cdot A\, |\, x \in \mathbb{F}_{q}^k\}\subset \mathbb{F}_{q}^n$的生成矩阵的转置这一性质作为此类实例难度的判定准则，并证明以下下界：1) 对于Res(lin$_{\mathbb{F}_q}$)的有向无环（dag-like）片段。我们引入子集和实例的$(s,r)$-鲁棒性概念，其蕴含矩阵$A$定义了一个最小距离$s\geq r$的纠错码。对于$(s,r)$-鲁棒实例，我们证明了Res(lin$_{\mathbb{F}_q}$)有向无环片段中驳斥规模的下界为$2^{Ω(r)}$。我们表明随机实例是$(n / 3, Ω\left((n/(q + 1)\ln q))^{1/3}\right))$-鲁棒的，且可通过代数几何码构造达到这些界的特例。2) 对于树状Res(lin$_{\mathbb{F}_q}$)驳斥，我们证明任意子集和实例的规模下界为$2^{Ω({((q+1)\ln q)^{-1/3}}d^{1/5})}$，其中$d$是码$C_{A}$的最小距离。