We obtain an explicit criterion for $p$-ary functions to produce association schemes in terms of their Walsh spectrum. Employing this characterization, we explicitly find a correlation between $p$-ary bent functions and association schemes; to be more exact, we prove that a $p$-ary bent function induces a $p$-class association scheme if and only if the function is weakly regular. As applications of our main criterion, we construct many infinite families of few-class association schemes arising from $p$-ary functions. Furthermore, we present four classes of $p$-ary two-weight linear codes, which are constructed from the association schemes produced in this paper.

翻译：我们获得了按沃尔什频谱计算协会计划所需的美元职能的明确标准。我们利用这一特征,明确发现美元弯曲职能和协会计划之间的相互关系;更确切地说,我们证明,美元弯曲函数只有在功能不固定的情况下,才会产生美元平方圆协会计划。作为我们主要标准的应用,我们建造了许多由美元平方圆功能产生的由几级协会计划组成的无限家庭。此外,我们提出了四类美元平方圆双重量线性代码,这些代码是根据本文件所制定的协会计划构建的。