22 problems

Helleseth–Kholosha conjecture. The Coulter–Matthews bent functions $f_a(x)=\operatorname{Tr}(ax^d)$ are weakly regular bent.

Mesnager's conjecture. For every even $m$, the function $f_{a,b}$ is bent if and only if

A Boolean function in an even number of variables is homogeneous if all monomials in its algebraic normal form have the same degree; it is rotation symmetric if it is invariant und…

Let $q=2^e$ with $e$ an even positive integer, let $u$ be a generator of $\mu_{q+1}$, and for $\alpha\in{\mathbb F}_{q}^*$ define $g_\alpha$ on ${\mathbb F}_{q^2}$ by…

Let $q=2^e$ with $e$ an even positive integer, let ${\bb F}_{q^2}$ be the finite field of order $q^2$, and for each $\alpha\in{\bb F}_{q}^*$ define … where $\sigma(X)=X+X^d+X^{dq}$…

Let $p$ be an odd prime. Let $F$ be the function defined by Eq. (10), with $t=m$, and let $b\geq 2$ satisfy $b\mid(p^m-1)$. Define … and let $C_{D_{F,H_b}}$ be the linear code defi…

Tokareva–Shapovalov conjecture. The function $f$ is a derivative of a bent function.

Tokareva's conjecture. Every such function $f$ is a derivative of some bent function.

The automorphism-group characterization conjecture. Quadratic Boolean bent functions can be characterized by 2-transitivity of one of the following automorphism groups:

Let $f$ be a bent function on $\mathbb F_2^n$. Write its Smith normal form as … Here $d_i$ are the elementary divisors and $m_i$ their multiplicities. Symmetry conjecture. The elem…

Let $n=2k>8$, let $F(x)=x^{d(2^k+1)}$ over $\mathbb{F}_{2^n}$, and assume ${\rm gcd}(d,2^k-1)=1$. For $\alpha\in\mathbb{F}_{2^n}\setminus\mathbb{F}_{2^k}$, write…

EA-equivalence conjecture. The designs ${\mathbb D}(f_1,\ldots,f_\ell)$ and ${\mathbb D}(g_1,\ldots,g_\ell)$ are isomorphic if and only if the vectorial Boolean functions…

Exponential-growth conjecture. For any given $\ell$ in the range $1\leq \ell\leq m$, the number of inequivalent codes with these parameters obtained from $(2m,\ell)$ bent vectorial…

Conjectural Gauss-sum formula. There exists a Boolean function $h_{a,b}(omega)$ such that

Bentness conjecture. The function $f_{a,b}$ is bent if and only if $K_{n/2}(a)=4$.

Let $\varDelta_m$ $(m\geqslant 1)$ be the sequence of edge-coloured graphs, each with two edge colours, constructed in the cited work. Colour-swapping automorphism conjecture. For…

Let $f:GF(p)^n\to GF(p)$ be an even bent function with $p>2$ and $f(0)=0$. For each value of $f$, its level curve is the corresponding level set; assume these level curves give ris…

Let $p>2$ and let $f:GF(p)^n\to GF(p)$ be a weakly regular bent function corresponding to a weighted strongly regular graph via the stated analogy. Let $\mu_{ii}$ denote the diagon…

A Boolean function is homogeneous if all its monomials have the same degree, and rotation symmetric if it is invariant under cyclic permutations of its variables. A Boolean functio…

A strongly regular graph (SRG) is a graph with constant parameters $(v,k,\lambda,\mu)$, and a graph is of negative Latin square type when its parameters have that form. The affine…

Let $p$ be a prime, let $q>1$ be a power of $p$, and let $n>1$. For $a\in\mathbb{F}_{q^n}^*$, define the Kloosterman sum … where $\operatorname{Tr}:\mathbb{F}_{q^n}\to\mathbb{F}_p$…

Let $n,k\geq 1$ be integers with $\gcd(k,n)=1$, and let $a\in\mathbb F_{3^n}^*$. The function … is a ternary bent function, called a Coulter–Matthews bent function. Weak regularity…