On minimal distance between q-ary bent functions

TL;DR

This paper establishes the minimal Hamming distance between distinct p-ary bent functions and counts the number of such functions at this minimal distance from a quadratic bent function, advancing understanding of their structure.

Contribution

It proves the minimal Hamming distance between p-ary bent functions and enumerates functions at this distance from a quadratic bent function, providing new insights into their combinatorial properties.

Findings

Minimal Hamming distance between p-ary bent functions is p^n.

Number of p-ary bent functions at this distance from quadratic bent function is p^n(p^{n-1}+1)...(p+1)(p-1).

Results hold for any prime p and even number of variables 2n.

Abstract

The minimal Hamming distance between distinct $p$-ary bent functions of $2n$ variables is proved to be $p^n$ for any prime $p$. It is shown that the number of $p$-ary bent functions at the distance $p^n$ from the quadratic bent function is equal to $p^n(p^{n-1}+1)\cdots(p+1)(p-1)$ as $p>2$.