AI 中文总结
研究如何通过阿廷 - 施赖埃尔多项式生成完全等分布的拟蒙特卡罗序列，结合索博尔构造与帕斯卡矩阵张量幂等方法，明确保证特定多项式\(t = 0\)的条件，提供优化\((2b - 1)\)维组合\(t\)值的快速贪心算法。
AI 中文摘要
为了对函数进行数值积分，可以采用拟蒙特卡罗估计器，它在伪随机且分布良好的均匀采样位置对被积函数值求平均。更好的均匀性可改善最坏情况下的积分误差界。均匀性的一个标准度量由整数\(t\)值给出，其中\(t = 0\)时均匀性最佳。通过索博尔递归构造可以生成具有有界\(t\)值的样本序列，该构造使用不可约多项式的系数。虽然在伽罗瓦域\(\mathrm{GF}(b)\)上取\(b\)个一次多项式可得到\(t = 0\)的\(b\)维序列，但我们展示了保证特定高阶多项式\(t = 0\)的条件。特别是，当所选多项式仅相差一个常数时，我们将索博尔构造与帕斯卡矩阵的张量幂相关联，并展示了在这种情况下保证\(t = 0\)的简单条件。然后我们专注于阿廷 - 施赖埃尔不可约多项式，形式为\(p_i(x) = x^b - x + c_i\)，其中\(i \in \{1, \dots, b - 1\}\)且\(b\)是素数，我们明确了在\(b - 1\)维中始终保证\(t = 0\)的条件。结合\(b\)维一次索博尔序列和我们的\(b\)次\((b - l)\)维阿廷 - 施赖埃尔序列，我们提供了一种快速贪心算法，可优化\((2b - 1)\)维组合\(t\)值，同时保证在子空间中\(t = 0\)投影。
英文摘要
To numerically integrate a function, one may resort to Quasi-Monte Carlo estimators, that average integrand values at pseudo-random well-distributed uniform sampling locations. Better uniformity improves the worst-case integration-error bound. A standard measure of uniformity is given by an integer $t$ value, where $t=0$ yields the best uniformity. Producing sequences of samples with bounded $t$ values can be achieved with Sobol' recursive construction, that uses coefficients of irreducible polynomials. While $b$-dimensional sequences with $t=0$ can be obtained by taking $b$ polynomials of degree $1$ over the Galois Field $\mathrm{GF}(b)$, we show conditions that guarantee $t=0$ for specific higher degree polynomials. In particular, we relate the Sobol' construction to tensorized powers of Pascal matrices when the chosen polynomials only differ by a constant and exhibit simple conditions to guarantee $t=0$ in this case. We then focus on Artin-Schreier irreducible polynomials, in the form $p_i(x) = x^b - x + c_i$, where $i \in \{1, \dots, b-1\}$ and $b$ is prime, and we make explicit conditions that always guarantees $t=0$ in $b-1$ dimensions. Combining $b$-dimensional Sobol' of degree $1$ and our $(b-1)$-dimensional Artin-Schreier sequence of degree $b$, we provide a fast greedy procedure that optimizes the $(2b-1)$-dimensional combined $t$ value, while guaranteeing $t=0$ projection in subspaces.
