Two elements g and h of a permutation group G acting on a set V are said to be intersecting if $g(v)=h(v)$ for some $v\in V$. More generally, a subset $\mathcal{F}$ of G is an intersecting set if every pair of elements of $\mathcal{F}$ is intersecting. The intersection density $\rho (G)$ of a transitive permutation group G is the maximum value of the quotient $|\mathcal{F}|/|G_v|$ where $G_v$ is the stabilizer of $v\in V$ and $\mathcal{F}$ runs over all intersecting sets in G. Intersection densities of transitive groups of degree pq, where $p>q$ are odd primes, is considered. In particular, the conjecture that the intersection density of every such group is equal to 1 (posed in Meagher et al. (2021) [15]) is disproved by constructing a family of imprimitive permutation groups of degree pq (with blocks of size q), where $p=(q^k-1)/(q-1)$, whose intersection density is equal to q. The construction depends heavily on certain equidistant cyclic codes ${[p,k]}_q$ over the field ${\mathbb{F}}_q$ whose codewords have Hamming weight strictly smaller than p.

作用在集合V上的置换群G 的两个元素g和h被称为相交，如果$G(v)=H(v)$ 对于一些 $v\in 伏$. 更一般地，一个子集$\mathcal{F}$的G ^是一个交叉组，如果每对元件的$\mathcal{F}$相交。的交叉点密度 $\rho (G)$传递置换群G是商的最大值$|\mathcal{F}|/|G_v|$ 在哪里 $G_v$ 是稳定器 $v\in 伏$ 和 $\mathcal{F}$遍历G 中的所有相交集。度为pq的传递群的交集密度，其中$p>q$是奇素数，被考虑。特别是，通过构造一个度为pq的非原始置换群族（具有大小为q）， 在哪里$p=(q^{克}-1)/(q-1)$，其交点密度等于q。构造在很大程度上依赖于某些等距循环码${[p,克]}_q$ 在场上 ${\mathbb{F}}_q$其码字的汉明权重严格小于p。