Let A be a random m × n matrix over the finite field $$\mathbb{F}_q$$ F q with precisely k non-zero entries per row and let $$y\in\mathbb{F}_q^m$$ y ∈ F q m be a random vector chosen independently of A . We identify the threshold m / n up to which the linear system A x = y has a solution with high probability and analyse the geometry of the set of solutions. In the special case q = 2, known as the random k -XORSAT problem, the threshold was determined by [Dubois and Mandler 2002 for k = 3, Pittel and Sorkin 2016 for k > 3], and the proof technique was subsequently extended to the cases q = 3,4 [Falke and Goerdt 2012]. But the argument depends on technically demanding second moment calculations that do not generalise to q > 4. Here we approach the problem from the viewpoint of a decoding task, which leads to a transparent combinatorial proof.

中文翻译：

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随机线性方程的可满足性阈值

设 A 是有限域 $$\mathbb{F}_q$$ F q 上的随机 m × n 矩阵，每行正好有 k 个非零项，并令 $$y\in\mathbb{F}_q^m$ $ y ∈ F qm 是一个独立于 A 选择的随机向量。我们确定线性系统 A x = y 具有高概率解的阈值 m / n 并分析解集的几何形状。在特殊情况 q = 2 中，称为随机 k -XORSAT 问题，阈值由 [Dubois and Mandler 2002 for k = 3, Pittel and Sorkin 2016 for k > 3] 确定，证明技术随后扩展到案例 q = 3,4 [Falke 和 Goerdt 2012]。但论点取决于技术上要求的二阶矩计算，这些计算不能推广到 q > 4。这里我们从解码任务的角度来解决问题，这导致了一个透明的组合证明。