29 problems

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 $n$ 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…

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 $m\geq 3$. 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…

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 $B_{n,k,\ell,q}$ 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 $C_O$ be the code associated with the Desarguesian ovoid in $Q^+(7,q)$, with $q\geq 3$. A code is strongly optimal when it is both $P$-optimal and $S$-optimal. Strong optimalit…

Let $D=\Delta_{M_1}+\eta\Delta_{M_2}+\cdots+\eta^{n-1}\Delta_{M_n}\subset\mathbb{F}_{2^n}^m$ be as above, with the hypotheses stated in the source, and let $C^{(2)}_{D^c}$ be the a…

Let $\mathbb{F}_{2^n}=\mathbb{F}_{2}(\eta)$ with $\eta\in\mathbb{F}_{2^n}$, and let … where $m\in\mathbb{N}$, $\emptyset\neq M_i\subsetneq[m]$ for $1\leq i\leq n$, and…

Let $\mathbb{F}_{2^n}=\mathbb{F}_{2}(\eta)$, let $m\in\mathbb{N}$, and let $\emptyset\neq M\subsetneq[m]$. Set … Let $C_{D^{\ast}}$ be the associated punctured simplicial code, and…

Let $m\ge 5$ be an odd integer. Let $f$ be any trinomial specified in the paper, let $D(f)^*$ be its punctured value-set, and let $\mathcal{C}_{D(f)^*}$ denote the associated binar…

Let ${\mathbb{D}}$ be the design defined earlier in the paper, and let ${\mathsf{C}}_p({\mathbb{D}})^\perp$ denote the dual of its code over the field of characteristic $p$. Dual m…

Let $q=2^m$, with $m\equiv 3\pmod 4$ and $m\geq 5$, and define … Let ${\mathcal{C}}_{(f,q)}^{(2)}$ be the associated binary subfield code. Nine-weight conjecture. The code…

Let $q=2^m$, and let ${\mathcal{C}}_{(f,q)}^{(2)}$ be the binary subfield code defined by the trace representation for an oval polynomial $f$. Nine-weight conjecture. For odd…

Let $m\ge 3$ be an odd integer, let $\alpha$ be a primitive element of $\operatorname{GF}(3^m)$, and let $\mathcal C_1$ and $\mathcal C_2$ be the ternary linear codes defined above…

Let $m\ge 3$ be an odd integer, let $\alpha$ be a primitive element of $\operatorname{GF}(3^m)$, and let $\mathcal C_1$ and $\mathcal C_2$ be the ternary linear codes defined above…

Let $m\ge 3$ be an odd integer, let $\alpha$ be a primitive element of $\operatorname{GF}(3^m)$, and let $\mathcal C_1$ and $\mathcal C_2$ be the ternary linear codes defined by ……

Let $q$ be a prime power, let $0<p<1-1/q$, and let $H_q(p)$ denote the $q$-ary entropy. A random linear code of rate $1-H_q(p)-\delta$ is chosen uniformly among linear codes of tha…

Let $ell \leq m$ be positive integers and let $t$ be an integer satisfying $1<t<ell$. For the determinantal code, write $hat{w}_r(t;ell,m)$ for the weight indexed by $r$. Weight-ord…

Let $\ell_q(r,3)$ be the length function, let $\ell_q(r,3,5)$ be the $d$-length function, let $s_q(r-1,2)$ be the smallest size of a $2$-saturating set in $\mathrm{PG}(r-1,q)$, and…

Let $m\geq 3$ be odd, let $v=(3^m-1)/2$, and let $\mathcal{C}_m$ be either of the ternary cyclic codes described in the paper. Let $A_k$ denote the number of codewords of Hamming w…

Ding's second difference-set conjecture. For the following three polynomials, with the indicated parity conditions,

J171 code conjecture. The binary code ${\mathcal C}_{D_f}$ has parameters