Nonexistence results on generalized bent functions Zqm → Zq with odd m and q 2 (mod 4)

Let p be an odd prime, let a be a positive integer, let m be an odd positive integer, and suppose that a generalized bent function from Z(2pa)(m) to Z(2pn) exists. We show that this implies m not equal 1, p <= 2(2m) + 2(m) + 1, and ord(p) (2) <= 2(m-1). We obtain further necessary conditions and prove that p = 7 if m = 3 and p is an element of {7,23,31, 73, 89} if m = 5. Our results are based on new tools for the investigation of cyclotomic integers of prescribed complex modulus, including "minimal aliases" invariant under automorphisms, and bounds on the l(2)-norms of their coefficient vectors. These methods have further applications, for instance, to relative difference sets, circulant Butson matrices, and other kinds of bent functions. (C) 2018 Elsevier Inc. All rights reserved.