64 problems

Incidence-code parameter conjecture. The paper conjectures the corresponding general formulas for the linear codes generated by incidence matrices of…

Connectivity and diameter conjecture. If both $m$ and $n$ are odd, then $G(\mathbb{Z}_{n}\oplus\mathbb{Z}_{m})$ is connected and

Let $r_F(n;s,t)$ denote the minimum redundancy of a binary functional $(s,t)$-batch code encoding $n$ information symbols. For $1\leqslant s\leqslant t\leqslant 2^n-1$, write…

Let $m\geq 5$ be odd, let $D_f$ be defined from one of the eleven trinomials in Ding's difference-set conjecture, and let ${\mathcal C}_{D_f}$ be the associated binary linear code.…

Let $C\subseteq\mathbb{F}_q^n$ be a random linear code of dimension $k$, and let its cube code be $C^{(3)}=C\ast C\ast C$, where $\ast$ denotes the Schur product of linear codes. C…

A locally leakage-resilient linear threshold secret sharing scheme has reconstruction threshold $t$ among its parameters. Bounded-threshold conjecture. There exist locally leakage-…

Let $v\geq 0$ and $\mathcal{T}\geq 1$, and consider binary polycyclic codes associated with … Here, an LCD code is a binary linear code whose intersection with its Euclidean dual i…

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 $X$ and $Y$ be binary random variables with $Y=X\oplus Z$, where under hypothesis $\mathcal{H}=i$, $Z\sim\operatorname{Bernoulli}(p_i)$ independently of $X$, and let the correl…

Let $C$ be a linear binary indel code of dimension $3$. Cheng et al.'s conjecture. There exists an absolute constant $\delta_0>0$ such that $C$ cannot correct a $1/2-\delta_0$ frac…

Let $M$ range over the matrices considered in the paper, and let $m(r)$ be the rank-dependent quantity defined by … For $r=1,\dots,n+1$, consider the maximum of $\theta_M$ among ma…

Let $n$ be a natural number, let $G(\mathbb{Z}_n)$ be a unit graph, and let $H$ be a $|V|\times|E|$ incidence matrix of $G(\mathbb{Z}_n)$. Let $\phi(n)$ denote Euler's totient func…

LMS conjecture. The lower bound $L\geq \ell^{\Omega(1/\eps)}$ applies to all linear codes achieving the Elias Bound for List Recovery.

Let $\mathcal{C}\subseteq\mathbb{F}_2^n$ be a linear code. A hitting set for $\mathcal{C}$ is a set $S\subseteq[n]$ such that $c_S\neq0$ for every nonzero codeword…

Let $n_1,n_2,r_1,r_2,k,\ell$ be positive integers with $n_1<r_1<2n_1$, and let $f_1(x),\ldots,f_\ell(x)$ be irreducible polynomials of degree $n_1n_2$ and exponent $r_1r_2$. Suppos…

Let $q$ be odd, let $t$ be odd, and for $0\leq k\leq m$ and $\delta\in\mathbb F_q$ define … where $G=\operatorname{diag}(1,1,\dots,1,\delta,0,\dots,0)$ has rank $k$. Minimum-weight…

Let $G$ be a generator matrix in standard form for a linear code over a field $F$. A code is linear complementary equi-dual (LCED) when it has a permutation-equivalent dual forming…

A linear code $C$ of length $n$ over a field $F$ is linear complementary equi-dual (LCED) if there is a code $D$ over $F$ such that $D$ is permutation equivalent to $C^{\perp}$ and…

Let $m\geq 3$. For the binary lifted Reed–Muller code ${\mathrm{RM}}_2(1,m)(2|2^2)$, let ${\mathbb{D}}_{3\times 2^{m-2}}({\mathrm{RM}}_2(1,m)(2|2^2))$ denote the design whose block…

Shi et al.'s conjecture. The largest number of non-zero weights of $\mathcal{C}$ is bounded above by

Let $N$ be a prime with $N\geq 3$. A spanning simplex is a lattice simplex whose lattice points at height one generate the ambient lattice, and a non-degenerate thin code is a thin…

Let $p>2$ be a prime, let $n$ be a positive integer, and let $f:\mathbb{F}_p^n\to\mathbb{F}_p$ be a function. Its Walsh transform at $\nu\in\mathbb{F}_p^n$ is … where…

Let $B_{n,k,\ell,q}$ be the number of inequivalent linear codes of length $n$, dimension $k$, and hull dimension $\ell$ over $\mathbb{F}_q$. Hull-dimension monotonicity hypothesis.…

Duality conjecture. A code $\mathcal{C}$ is recovery balanced if and only if its dual code $\mathcal{C}^\perp$ is recovery balanced.

Let $q$ be a prime power, let $b$ be a positive integer, and let $t$ be a positive integer. Let $k=t b$, and let ${\mathcal C}$ be an $(n,k,d_b)^b_q$ cyclic code, where $d_b$ denot…