904 problems

Let $n$, $r$, and $q$ be positive integers, and let a $q$-ary perfect $r$-error-correcting Lee code of length $n$ be a perfect $r$-error-correcting Lee code over an alphabet of siz…

Belfiore–Solé conjecture. For every unimodular lattice $L$, the function $\Xi_L(y)$ attains its global maximum on the positive imaginary axis at the symmetry point $y=1$.

Let $\mathcal{C}$ be an $[n,k]$ MDS code over the finite field $\mathbb{F}_q$. Segre's MDS conjecture. Then … This conjecture concerns the maximum possible length of MDS codes over…

Etzion–Silberstein conjecture. For every Ferrers diagram $\mathcal{F}$ of order $n$, every $1\le d\le n$, and every finite field $\mathbb{F}_q$, there exists an…

An irreducible cyclic code is denoted by $\mathcal{C}(k,q)$, and a code is 2-weight when its nonzero codewords have exactly two distinct Hamming weights. The semiprimitive codes, s…

Simplex-code optimality conjecture. Then $\mathcal{C}$ solves Problem. The paper reports experimental evidence that simplex codes perform best among all codes with the same paramet…

Shi et al.'s conjecture. The bound is tight for all values of $q$ and $k$.

Delsarte's conjecture. There are no nontrivial perfect codes in Johnson graphs.

A $t$-design is a combinatorial design whose every set of $t$ points lies in the same number of blocks. For a prime power $q=p^s$, a geometric design is the design formed by all…

Let $d,m$ be positive integers, let $S_d$ be the space of homogeneous degree-$d$ polynomials in $m+1$ variables over ${\mathbb F}_q$, and let $e_r(d,m)$ be the maximum number of…

An optimal recursive tower is a recursive tower of algebraic function fields attaining the Drinfeld–Vlăduț bound. Elkies' conjecture. Every optimal recursive tower is modular in an…

Let $\mathcal{P}$ be a binary-input memoryless symmetric (BMS) channel with output alphabet $\mathcal{Y}$, and let … be its capacity under the uniform input distribution. For a seq…

Neumaier's conjecture. The only completely regular code containing more than two codewords with $d\geq 8$ is the extended binary Golay code.

Let terminated spatially-coupled low-density parity-check (SC-LDPC) codes be constructed from an underlying LDPC code ensemble, and let their BP threshold and the MAP threshold of…

Let $M(k,q)$ denote the maximal length of a non-trivial $q$-ary MDS code of dimension $k$. For $q>k$, the Main Conjecture on MDS codes. … This is a central parameter problem for MD…

Charpin's conjecture. Almost all cyclic codes are standard.

Given a binary sequence $\mathbf{b}=(b_j)_{j=0}^{n-1}$ with $b_j\in\{\pm1\}$, let $F$ denote its merit factor. Barker-sequence merit-factor conjecture. The values $169/12$ and…

Let $q$ be a prime power, let $\theta$ specify the twist defining the twisted Reed–Solomon code ${\rm TRS}_k(\mathbb{F}_q,\theta)$, and let $\mathcal{S}_{k,\theta}$ be the source-d…

In a memory approximate message passing (MAMP) receiver, let $\hat{\phi}_t(\cdot)$ denote the code constraint associated with the forward-error-correction (FEC) decoder at iteratio…

Let $\mathbb{P}_2(n)$ denote the lattice of subspaces of an $n$-dimensional vector space over $\mathbb{F}_2$. A subset $\mathcal{U} \subseteq \mathbb{P}_2(n)$ is a linear code if i…

Tsfasman's conjecture. If $d\leq q+1$, then

Ferrers diagram rank-metric code existence conjecture. For any $q,m,n,k$, there always exists a Ferrers diagram rank-metric code attaining this bound. This is important because cod…

Let $\rho_q(n)$ be the maximum nonlinearity of a function from $\mathbb{F}_q^n$ to $\mathbb{F}_q$, and define … In particular, $mu_2(n)$ denotes this quantity for Boolean functions…

Goppa's conjecture. The Gilbert–Varshamov bound is tight in the binary case; equivalently, the asymptotic achievable rate equals the Gilbert–Varshamov bound.

Let $\mathbb{F}_q$ be a finite field, let $D=\mathbb{F}_q^*$ be the evaluation set of a primitive Reed–Solomon code with code length $n=q-1$, dimension $k$, and codebook…