We first find an explicit criterion for a p-ary function to produce a symmetric association scheme. As the next step, we proceed to apply this criterion to the p-ary plateaued functions of \(\ell \) -form. We thus obtain a necessary and sufficient condition for the p-ary plateaued functions of \(\ell \) -form to yield symmetric association schemes; in fact, this condition is that f is weakly regular partially bent. For the proof of our main results, we use the equivalent characterizations for the gcd-condition of p-ary plateaued functions of \(\ell \) -form (including non-weakly regular bent functions). Using this characterization, we also find some sufficient conditions for p-ary plateaued functions of \(\ell \) -form to satisfy the gcd-condition. In particular, we show that the gcd-condition holds for any non-weakly regular bent function of \(\ell \) -form. We further provide explicit constructions for two families of 3-class and 4-class association schemes using our main results.
