标题： 广义三差问题 显示英文标题

标题： The generalized trifference problem

摘要： 我们研究了寻找最长为$n$的三元向量的最大数量$T(n, m)$的问题，使得对于任意三个不同的向量，在它们两两之间至少有$m$个坐标位置不同。 对于$m = 1$，这是经典的三差问题，目前仍未解决。 我们证明了参数$m$的不同范围内$T(n, m)$的上下界，并确定了$m=m(n)$上相变阈值，在该阈值处$T(n, m)$从常数跳跃到指数级别$n$。 通过将该问题的线性版本与有限几何中的阻塞集问题相关联，我们给出了明确的构造和概率下界。 我们也计算了小参数下该函数及其线性变化的确切值。 显示英文摘要

摘要： We study the problem of finding the largest number $T(n, m)$ of ternary vectors of length $n$ such that for any three distinct vectors there are at least $m$ coordinates where they pairwise differ. For $m = 1$, this is the classical trifference problem which is wide open. We prove upper and lower bounds on $T(n, m)$ for various ranges of the parameter $m$ and determine the phase transition threshold on $m=m(n)$ where $T(n, m)$ jumps from constant to exponential in $n$. By relating the linear version of this problem to a problem on blocking sets in finite geometry, we give explicit constructions and probabilistic lower bounds. We also compute the exact values of this function and its linear variation for small parameters.