# On $q$-nearly bent Boolean functions

## TL;DR

This paper introduces the concept of $q$-nearly bent Boolean functions, proving their existence for certain classes and providing conditions to identify non-$q$-nearly bent functions, advancing understanding of Boolean function transformations.

## Contribution

It establishes the existence of $q$-nearly bent functions for non-affine $q$ with weight one, answering an open question and providing criteria for non-$q$-nearly bent functions.

## Key findings

- Balanced Boolean functions are $q$-nearly bent if $q$ has weight one.
- Proved existence of non-affine $q$-nearly bent functions.
- Provided a necessary condition to determine when a function isn't $q$-nearly bent.

## Abstract

For each non-constant Boolean function $q$, Klapper introduced the notion of $q$-transforms of Boolean functions. The {\em $q$-transform} of a Boolean function $f$ is related to the Hamming distances from $f$ to the functions obtainable from $q$ by nonsingular linear change of basis. In this work we discuss the existence of $q$-nearly bent functions, a new family of Boolean functions characterized by the $q$-transform. Let $q$ be a non-affine Boolean function. We prove that any balanced Boolean functions (linear or non-linear) are $q$-nearly bent if $q$ has weight one, which gives a positive answer to an open question (whether there exist non-affine $q$-nearly bent functions) proposed by Klapper. We also prove a necessary condition for checking when a function isn't $q$-nearly bent.