289 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 $k=3$…

Consider the real additive white Gaussian noise sequence channel … where the noise coordinates are independent $N(0,N_0/2)$. An $(E,M,\epsilon)$ code has $M$ equiprobable messages,…

Let $P(v_1,\ldots,v_N)=\int_{\mathbb R^n}\max_i e^{-|x-v_i|^2}\,dx$ for a code consisting of $N$ vectors in $\mathbb R^n$. The simplex conjecture. For $N=n+1$ and fixed…

For every integer $n\ge 1$, energy $E>0$, and noise standard deviation $\sigma>0$, let $x_0,\ldots,x_n\in\mathbb{R}^n$ satisfy $\lVert x_i\rVert^2=E$ for all $i$, and let the signa…

Power-weight duality conjecture. The power weights $w^{(\ell)}$, $\ell \geq 2$, do not respect duality over any $\mathbb{Z}/m\mathbb{Z}$ with $m \geq 4$.

Let $n$ be the alphabet length, let $s_r$ denote the size of the sphere of radius $r$, and let $\binom{n}{r}$ be the binomial coefficient. The quantities $\frac{s_r}{\binom{n}{r}}$…

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…

Conjecture on the 2-rank of $M_2$.

Odd-cycle naive-dimension conjecture. The cyclic interval lemma holds for every odd $m$; consequently,

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 linear code $\mathcal{C}\subseteq\mathbb{F}_q^n$ of dimension $k$ over every finite field $\mathbb{F}_q$, and every admissible order $t$ with $1\le t\le k$, the $t$-th ge…

Let $\chi_{\mathrm{lat}}(\mathbb{R}^n)$ denote the minimum number of colors in a lattice coloring of $\mathbb{R}^n$ such that no two points of the same color are at Euclidean dista…

Let an $\varepsilon$-synchronization string of length $n$ over an alphabet $\Sigma$ be a word $S\in\Sigma^n$ such that, for every $0\le i<j<k\le n$, the insertion-deletion edit dis…

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…

Let $G_q$ be an alphabet of size $q\ge 2$, let $t\ge 1$ be fixed, and let $C\subseteq G_q^n$. For $X,Y\in G_q^{t\times n}$, define the column Hamming distance by…

Let $X^n=(X_1,5e5c,5e5c,X_n)$ have independent coordinates distributed as $\operatorname{Bernoulli}(p)$, with $0<p<\tfrac12$, and let distortion be normalized Hamming distortion. D…

For every pair of integers $n,k$ satisfying $3\leq k\leq n-1$, the optimal leading coefficient in the high-SNR maximum-likelihood error-probability expansion for SNR-dependent…

For every integer $k\ge 2$, let $N=2^k-1$. For every integer $t$ with $1\le t\le N-1$, the number of pairs $(a,b)\in\{0,1,\ldots,N-1\}^2$ satisfying $a+b\equiv t\pmod N$ and…

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…

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Let $\mathcal{P}_{d}({\mathbb{F}}^n)$ be the class of polynomials of degree at most $d$ from ${\mathbb{F}}^n$ to ${\mathbb{F}}$. For a function $g:{\mathbb{F}}^n\to{\mathbb{F}}$, l…

Let $n$ be a composite positive integer, let $q$ be a prime power with $\operatorname{char}(\mathbb{F}_q)\nmid n$, and let $\mathcal{C}_{n,1}$ be the cyclic code over…

Let $w=(w_0,w_1,ots,w_m)\in\mathbb{N}_{\geq 1}^{m+1}$, let $\mathbb{P}(w)$ be the weighted projective space over $\mathbb{F}_q$, and let $e_q(d;w_0,w_1,\dots,w_m)$ denote the maxim…

Let $H_q^n$ be the Hamming space over an alphabet of size $q$, and let $C\subseteq H_q^n$ be an equidistant code with distance $d$. Hegedüs' conjecture. If … then … This conjecture…