Nonexistence of generalized bent functions and the quadratic norm form equations

Abstract

We present a new result on the nonexistence of generalized bent functions (GBFs) from \((\mathbb {Z}/t\mathbb {Z})^n\) to \(\mathbb {Z}/t\mathbb {Z}\) (called type [n, t]) for a large class. Assume p is an odd prime number. By showing certain quadratic norm form equations having no integral points, we obtain a universal result on the nonexistence of GBFs with type \([n, 2p^e]\) when p and n satisfy a certain inequality, and by computational methods with a widely accepted hypothesis, Generalized Riemann Hypothesis, we also achieve some results on the nonexistence of GBFs for relatively small p.