12 problems

Let $n>3$ be a prime, and let $\operatorname{ord}_n(3)$ denote the multiplicative order of $3$ modulo $n$. Set … Let $\mathcal N(n,3)$ denote the family of cyclic codes over…

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 $q$ be a fixed prime power. A family of cyclic codes over $\F_q$ is asymptotically good if it has positive limiting rate and positive limiting relative distance. Cyclic-code a…

Let $q$ be a fixed prime power. An infinite family of cyclic codes over $\F_q$ has constant relative distance if its relative distance is bounded below by a positive constant, and…

Let $m=2^l e>2$, where $2\nmid e$ and $l\geq1$, and let $\mathcal{C}_{\mathcal{D},i}$ denote the cyclic code associated with the sequence family $\mathcal{D}$. Let $h$ be the param…

Let $q=5^m$ with $m\geq3$ odd. Let $\mathcal C_{(q,q+1,4,1)}$ be the BCH code appearing in the source, and let $\overline{\mathcal C}_4$ be the extended classical MDS cyclic code.…

Let $q=p^s$ be a prime power, let $Q=q^m$, and let $N$ be a divisor of $Q-1$. Write $N_1=(Q-1)/N$ and let $A$ be the circulant matrix defined in equation from the relevant Gaussian…

Symmetry conjecture. If $v$ and $\mathantt{e}$ work together, then $v$ is symmetric.

Cyclic subspace code existence conjecture. There exists a cyclic subspace code

Let $q$ be a prime power and let $C^*=[q^2+1,q^2+1-d,d]_{q^2}$ be an $F_{q^2}$-linear (consta)cyclic MDS code. Let $P(C)$ denote its corresponding puncture code. Puncture-code dist…

Let $\mathcal{G}_{q}(n,k)$ denote the Grassmannian of $k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$, and let a cyclic code be a subspace code invariant under multiplication by…

Let $m=2t$, let ${\mathbb F}$ be the finite field of size $2^m$, and let ${\mathcal C}_d$ be the binary cyclic code of length $2^m-1$ and dimension $2m$ with two nonzeros…