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：关联方案在代数组合学中扮演重要角色，并在编码理论、图论和设计理论中具有重要应用。利用弯曲函数构造关联方案的方法已得到广泛研究。最近，在文献[13]中，Özbudak和Pelen利用三元非弱正则弯曲函数构造了无限族的5类和6类对称关联方案。他们还指出，从p元非弱正则弯曲函数构造2p类关联方案是一个有趣的问题，其中p>3为奇素数。本文利用非弱正则弯曲函数，对任意奇素数p构造了无限族的2p类、(2p+1)类和(3p+1)/2类对称关联方案。通过融合这些关联方案，我们还得到了t类对称关联方案，其中t=4,5,6,7。此外，我们给出了划分P、D、T、U和V（本文中定义）诱导对称关联方案的充分必要条件。