ABSTRACT |
A p-ary function f in n variables is an l-form if f (tu) = t(l) f (u) for any nonzero t in Z(p) and u in Z(p)(n). Let n be a positive even integer, p an odd prime, and l an element of {1, 2,..., p-1} provided that l not equal p-1 if p > 3. Let f be a p-ary bent function in n variables of l -form with f (0) = 0 and gcd(l -1, p -1) = 1, and let H-l = {t(l) : t is an element of Z(p)*}. We denote by G(f,) (l) the Cayley graph Cay(Z(p)(n), U-s is an element of Hl f(-1)(s)). Our main results are as follows: 1) if there is weakly regular p-ary bent f which is not regular, then l is 2; 2) if l = 2, then f is weakly regular p-ary bent if and only if the Cayley graph G(f, l) is strongly regular; 3) if l not equal 2, then f is regular p-ary bent if and only if the Cayley graph G(f, l) is strongly regular; 4) G(f, l) can be replaced by Cay(Z(p)(n), f(-1)(0)\{0}) in 2) and 3); and 5) amorphic association schemes are derived by using 2) and 3). We prove our main results by computing at most four distinct restricted eigenvalues of G(f, l). |