Cyclic bent functions and their applications in codes, codebooks, designs, MUBs and sequences

TL;DR

This paper introduces a new class of cyclic bent functions, explores their construction, and demonstrates their applications in quantum physics, cryptography, and communication systems.

Contribution

It constructs a new, inclusive class of cyclic bent functions and applies them to design MUBs, codebooks, sequences, and binary codes with specific properties.

Findings

New class of cyclic bent functions constructed

Applications in MUBs, codebooks, and sequences demonstrated

Binary codes including Kerdock code derived from cyclic bent functions

Abstract

Let $m$ be an even positive integer. A Boolean bent function $f$ on $\GF{m-1} \times \GF {}$ is called a \emph{cyclic bent function} if for any $a\neq b\in \GF {m-1}$ and $\epsilon \in \GF{}$, $f(ax_1,x_2)+f(bx_1,x_2+\epsilon)$ is always bent, where $x_1\in \GF {m-1}, x_2 \in \GF {}$. Cyclic bent functions look extremely rare. This paper focuses on cyclic bent functions on $\GF {m-1} \times \GF {}$ and their applications. The first objective of this paper is to construct a new class of cyclic bent functions, which includes all known constructions of cyclic bent functions as special cases. The second objective is to use cyclic bent functions to construct good mutually unbiased bases (MUBs), codebooks and sequence families. The third objective is to study cyclic semi-bent functions and their applications. The fourth objective is to present a family of binary codes containing the Kerdock code as a special case, and describe their support designs. The results of this paper show that cyclic bent functions and cyclic semi-bent functions have nice applications in several fields such as symmetric cryptography, quantum physics, compressed sensing and CDMA communication.