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

Abstract

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 Z2pam to Z2pa exists. We show that this implies m≠1, p≤22m+2m+1, and ordp(2)≤2m−1. We obtain further necessary conditions and prove that p=7 if m=3 and p∈{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 ℓ2-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.