论文标题
在强烈步行的常规图表上，三重总和及其代码
On strongly walk regular graphs, triple sum sets and their codes
论文作者
论文摘要
强烈的行走常规图（SWRGS或$ S $ -SWRG）形成了强烈规则图（SRG）的自然概括，其中长度〜2的路径被长度〜$ s $的路径所取代。它们可以被构造为投影三重代码双重二元组的cost图，其权重满足某个方程式。我们为中等大小代码长度提供了这些代码可行参数的分类。对于二进制案例，研究了这些代码的权重的分裂性，并显示了一些一般结果。众所周知，$ s $ -swrg最多具有4个不同的特征值$ k>θ_1>θ_2>θ_3$，而三重$（θ_1，θ_2，θ_3）$满足某些同质的多项式等级$ s -2 $ s -2 $（van dam dam dam dam dam，omiidi，omomidi，2013年）。该方程定义了平面代数曲线；我们使用算法算术几何形状中的方法来表明，对于$ s = 5 $和$ s = 7 $，只有明显的解决方案，我们猜测这是所有（奇数）$ s \ ge 9 $保持正确的。
Strongly walk regular graphs (SWRGs or $s$-SWRGs) form a natural generalization of strongly regular graphs (SRGs) where paths of length~2 are replaced by paths of length~$s$. They can be constructed as coset graphs of the duals of projective three-weight codes whose weights satisfy a certain equation. We provide classifications of the feasible parameters of these codes in the binary and ternary case for medium size code lengths. For the binary case, the divisibility of the weights of these codes is investigated and several general results are shown. It is known that an $s$-SWRG has at most 4 distinct eigenvalues $k > θ_1 > θ_2 > θ_3$, and that the triple $(θ_1, θ_2, θ_3)$ satisfies a certain homogeneous polynomial equation of degree $s - 2$ (Van Dam, Omidi, 2013). This equation defines a plane algebraic curve; we use methods from algorithmic arithmetic geometry to show that for $s = 5$ and $s = 7$, there are only the obvious solutions, and we conjecture this to remain true for all (odd) $s \ge 9$.
