280 problems

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 $p$, $m$, $r$, $s$, 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 $M_2$.

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 $\frac{s_r}{\binom{n}{r}}$…

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 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 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…

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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…

Fix a code ${\mathcal M}$. Given a code ${\mathcal C}$ of length $n$, polynomial-time minor-containment conjecture. It is decidable in time polynomial in $n$ whether ${\mathcal C}$…

Entropic Marton-type conjecture. There exists $c_2=\exp(O(1/c_1))$ such that, for every $m\in\mathbb{Z}_+$ and every such $X$, there is a subspace $G$ with

Deletion Bound I tightness conjecture. If set $\mathcal{J}$ satisfies both cover condition and swap condition, then Algorithm (Deletion Bound I) can find the exact value of…

Let $DPL(n,d)$ denote a diameter $d$ perfect Lee code in $\mathbb{Z}^{n}$. For odd $d$, this agrees with a perfect Lee code of radius $(d-1)/2$; for even $d$, it is defined using a…

Let $\mathbb{F}_2$ be the binary field and let $\mathbf{H}=\mathbb{F}_2^\Omega$. For $k\in[1,|\Omega|]$, write $\mathcal{P}(k,\Omega)$ for the covering of $\Omega$ by all $k$-eleme…

Let $A_q(n,d;k)$ denote the maximum size of a constant-dimension $k$-subspace code in $\mathbb{F}_q^n$ with minimum subspace distance at least $d$. Generalized vector space partiti…

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…

Decreasing normalized count conjecture. The sequence $\{a_k,\ k=10,11,12,\dots\}$ is decreasing. This conjecture concerns the growth of the number of inequivalent binary self-dual…

Let $B^{\sharp}$ be the domain of finite binary strings under consideration, and let $P^G(x)$ and $K^G(x)$ denote the corresponding generalized probability and complexity functions…

Let $B^{\sharp}$ be the domain of finite binary strings under consideration, and let $P^E(x)$ and $K^E(x)$ denote the corresponding enumerable measure and complexity functions. Wri…

Let $B^{\sharp}$ be the domain of finite binary strings under consideration, and let $P^M(x)$ and $K^M(x)$ denote the corresponding measure and complexity functions. Write…

Let ${\mathcal D}$ be a symmetric design with Singer parameters … where $q=p^s$ and $p$ is prime. Write ${\rm rank}_p{\mathcal D}$ for the dimension over ${\mathbb F}_p$ of the cod…

Let $p>5$ be an odd prime, let $C_{NQ}\subset {\mathbb F}^{2p}$ be the quasi-quadratic residue code, and let $C'$ be the code spanned by $C_{NQ}$ and the all-ones codeword. Let…

Let $p>5$ be an odd prime, let $Q,N\subset GF(p)^\times$ be the quadratic residues and non-residues, and let $C_{NQ}$ be the quasi-quadratic residue code … where…

For an odd prime $p$ and each non-empty subset $S\subset GF(p)$, let $X_S$ be the hyperelliptic curve defined by … Write $B(c,p)$ for the assertion that $|X_S(GF(p))|\leq c\,p$ for…