Let $A$ be a random $m\times n$ matrix over the finite field $F_q$ with precisely $k$ non-zero entries per row and let $y\in F_q^m$ 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, Dietzfelbinger et al. 2010, Pittel and Sorkin 2016], 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 $q>3$. Here we approach the problem from the viewpoint of a decoding task, which leads to a transparent combinatorial proof.
翻译:$A=y$在有限字段中是一个随机的$m\乘以nn美元矩阵,每行不折不扣不折不扣地输入美元,每行不折不扣地以美元为单位,让美元作为随机矢量。我们确定了线性系统以美元x=y$具有高概率解决方案的阈值m/n$,并分析了一套解决方案的几何。在被称为随机美元-XORSAT问题的特例中, $q=2美元, 门槛由[Dubois and Mandler 2002, Dietzfelbinger et al. 2010, Pittel and Sorkin 2016] 确定, 并随后将证明技术扩大到案件$q=340美元[Falke 和 Goerdt 2012]。但这一论点取决于技术要求的第二秒计算,但不会将美元整为$>3。我们在这里从解码任务的角度处理该问题,这将导致一个透明的组合证据。