Nonexistence of generalized bent functions and the quadratic norm form equations

TL;DR

This paper proves the nonexistence of certain generalized bent functions by analyzing quadratic norm form equations and employing computational methods under the Generalized Riemann Hypothesis, providing broad and specific nonexistence results.

Contribution

It introduces a new approach linking quadratic norm form equations to the nonexistence of GBFs, extending known results to larger classes and specific small cases.

Findings

Nonexistence of GBFs for type [n, 2p^e] under certain conditions

No integral solutions to specific quadratic norm form equations

Results supported by computational evidence under GRH

Abstract

We present a new result on the nonexistence of generalized bent functions (GBFs)from (Z/tZ)^n to Z/tZ (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 universalresult 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.