Abstract We consider the cyclic presentation of $$\textrm{PG}(3, q )$$ PG ( 3 , q ) whose points are in the finite field $$\mathbb {F}_{q^4}$$ F q 4 and describe the known ovoids therein. We revisit the set $$\mathcal {O}$$ O , consisting of $$(q^2+1)$$ ( q 2 + 1 ) th roots of unity in $$\mathbb {F}_{q^4}$$ F q 4 , and prove that it forms an elliptic quadric within the cyclic presentation of $$\textrm{PG}(3, q )$$ PG ( 3 , q ) . Additionally, following the work of Glauberman on Suzuki groups, we offer a new description of Suzuki–Tits ovoids in the cyclic presentation of $$\textrm{PG}(3, q )$$ PG ( 3 , q ) , characterizing them as the zeroes of a polynomial over $$\mathbb {F}_{q^4}$$ F q 4 .

Let m be an even positive integer. A Boolean bent function f on F(2m-1)×F2is called a cyclic bent function if for any a≠b∈F(2m-1) and ε∈F2, f( ax1,x2)+f( bx1,x2+ε) is always bent, where x1∈F(2m-1),x2∈F2. Cyclic bent functions look extremely rare. This paper focuses on cyclic bent functions on F(2m-1)×F2and their applications. The first objective of this paper is to establish a link between quadratic cyclic bent functions and a special type of prequasifields, and construct a class of quadratic cyclic bent functions from the Kantor-Williams prequasifields. The second objective is to use cyclic bent functions to construct families of optimal sequences. The results of this paper show that cyclic bent functions have nice applications in several fields such as coding theory, symmetric cryptography, and CDMA communication.