Association schemes play an important role in algebraic combinatorics and have important applications in coding theory, graph theory and design theory. The methods to construct association schemes by using bent functions have been extensively studied. Recently, in [13], {\"O}zbudak and Pelen constructed infinite families of symmetric association schemes of classes $5$ and $6$ by using ternary non-weakly regular bent functions.They also stated that constructing $2p$-class association schemes from $p$-ary non-weakly regular bent functions is an interesting problem, where $p>3$ is an odd prime. In this paper, using non-weakly regular bent functions, we construct infinite families of symmetric association schemes of classes $2p$, $(2p+1)$ and $\frac{3p+1}{2}$ for any odd prime $p$. Fusing those association schemes, we also obtain $t$-class symmetric association schemes, where $t=4,5,6,7$. In addition, we give the sufficient and necessary conditions for the partitions $P$, $D$, $T$, $U$ and $V$ (defined in this paper) to induce symmetric association schemes.

翻译：结合方案在代数组合论中扮演着重要角色，并在编码理论、图论和设计理论中具有重要应用。利用 bent 函数构造结合方案的方法已被广泛研究。近期，Özbudak 和 Pelen 在文献 [13] 中利用三元非弱正则 bent 函数构造了类数分别为 $5$ 和 $6$ 的无限族对称结合方案。他们指出，从 $p$ 元非弱正则 bent 函数构造 $2p$ 类结合方案是一个有趣的问题，其中 $p>3$ 为奇素数。本文利用非弱正则 bent 函数，对任意奇素数 $p$ 构造了类数分别为 $2p$、$(2p+1)$ 和 $\frac{3p+1}{2}$ 的无限族对称结合方案。通过融合这些结合方案，我们还获得了 $t$ 类对称结合方案，其中 $t=4,5,6,7$。此外，我们给出了划分 $P$、$D$、$T$、$U$ 和 $V$（定义于本文中）诱导对称结合方案的充要条件。