845 problems

Let $\mathbb{F}_q$ be a finite field, and let $[n,k]$ denote a linear code of length $n$ and dimension $k$ over $\mathbb{F}_q$. If $1<k<q$, then … except when $q$ is even and $k=3$…

Let $a_{11},a_{22},a_{33},a_{12},a_{13},a_{23},d$ be integers, and consider the generalized Markoff equation … Let $D$ and $A_{11},A_{22},A_{33}$ denote the quantities defined earl…

Let $p\geq 3$ be prime. A Hartke $\mathbb{Z}_p$-magic graph is a graph with the Hartke magic-labeling property over $\mathbb{Z}_p$; its order is its number of vertices. Low–Roberts…

Let $f\in\mathbb{Z}[x,y]$, let $C_p$ be the curve modulo $p$ defined by $f(x,y)\equiv0\pmod p$, and write $e_{p^2}(u)=\exp(2\pi i u/p^2)$. Exponential-sum conjecture. There exists…

Let $\mathbb{K}$ be a finite field of characteristic different from $2$, and let $\mathbb{K}^{*}$ denote its multiplicative group. An element $x$ is a generator if it generates…

Let $M_1=x^2+x+1$, and let the odd sequence of a polynomial be the sequence of odd polynomials obtained by the binary polynomial Collatz transformations. Let $r\geq 1$ and…

Let $p$, $m$, $r$, $s$, and $t$ be positive integers with $p$ prime, let $\boldsymbol{\alpha}=(\alpha_1,\ldots,\alpha_r)\in(\mathbb{F}_{p^m})^r$ have pairwise distinct entries, and…

Let $m=p_1\cdots p_r$ be a square-free product of $r$ distinct primes, and let $K_{n,d}^m$ denote the number of distinct zeros of a random polynomial in…

Let $q$ be an odd prime power and let $n\geq 3$. A quadratic system is a map … where each $f_i\in\mathbb{F}_q[x_1,\dots,x_n]$ has degree at most $2$. Classification conjecture. Suc…

Nonexistence conjecture. The polynomial $f(x)$ is not a permutation polynomial over $\mathbb{F}_{q^2}$.

Classification conjecture. A $\Phi_{\kappa}$-sequence $(a_n)_n$ is complete if and only if

Let $q$ be a prime power, let $\sigma$ be an operator with $\sigma^n=\operatorname{id}$, and let $W_1$ and $W_2$ be $\sigma$-invariant subspaces satisfying … Write $h_{\sigma}(V)$ for…

Let $q=p^m$ and $Q=p^e$. For integers $n$ and $\ell\geq 1$, let $g_{n,q}^{(\ell)}$ be the multivariate polynomial obtained from $g_{n,q}$ by composing with…

Prasad and Ram's conjecture. For any $T\in M_{3m}(\mathbb{F}_q)$,

Orbit-degree decomposition conjecture. Then $S_n$ splits as a direct sum of fields of degree $|\mathcal O_i|$ for all $i\in\{0,\dots,k\}$.

For every positive integer $n$, every positive integer $k$ with $\gcd(k,n)=1$, and every finite field $K$ of characteristic $2$ with $|K|=2^n$, let $d_k=4^k-2^k+1$ and…

For every integer $n\ge 6$ and every integer $k$ with $3\le k\le n-3$, there exists a $k$-dimensional $\mathbb{F}_2$-subspace $E\subseteq\mathbb{F}_{2^n}$ such that…

For every prime power $q>7$ and every $a_0\in\mathbb{F}_q$, there exists a circular permutation $(a_1,\ldots,a_{q-1})$ of $\mathbb{F}_q^{\times}$ such that, with $a_q=a_1$, each el…

For every nonsingular Hermitian variety $X$ of dimension $2m$ over a finite field, let $h$ denote the hyperplane class and let $N^m(X)_{\mathbb{R}}$ be the space of codimension-$m$…

For every prime power $q$, every integer $e\ge 2$, every integer $r\ge 1$, and every $a\in\mathbb{F}_{q^e}$, the binomial $f(X)=X^r\bigl(X^{q-1}+a\bigr)\in\mathbb{F}_{q^e}[X]$ is a…

For every prime power $q$ and every $p\in(0,1-1/q)$, let $H_q(p)$ denote the $q$-ary entropy. As $\varepsilon\to0$ with $1-H_q(p)-\varepsilon>0$, a random linear code…

Gow--McGuire Conjecture 1. Let $q\ne 13$ be any odd prime power. Let $\alpha\in\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$ be fixed. Then there exists $\lambda\in\mathbb{F}_{q}$ suc…

For every pair of positive integers $n,k$ with $\gcd(k,n)=1$, let $F:\mathbb{F}_{2^n}\to\mathbb{F}_{2^n}$ be defined by $F(x)=x^{4^k-2^k+1}$, and let…

For every prime power $q$ and integer $k\ge 0$, let $C_k(q)$ be the least cardinality of a family of nonzero polynomials over $\mathbb{F}_q$ whose associated codimension-$k$ partia…

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