27 problems

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

A two-weight cyclic code is a cyclic linear code having exactly two distinct nonzero Hamming weights among its codewords. Vega's conjecture. All two-weight cyclic codes are the "kn…

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

Let $n>3$ be a prime with $f=\operatorname{ord}_n(3)$ odd, and set $r=(n-1)/f$. Let $C_1,\ldots,C_r$ be the nonzero $3$-cyclotomic cosets modulo $n$. For a cyclic code $\mathcal C$…

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…

Essential-tightness conjecture. The bound in Theorem is essentially tight. The computed values for small BCH codes are consistent with this claim, but no matching lower bound is pr…

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 $m=2^l e>2$, where $2\nmid e$ and $l\geq1$, and let $\mathcal{C}_{\mathcal{S},i}$ denote the cyclic code generated by $M_{\mathcal{S},i}(x)$. Define … Here $d(\mathcal{C})$ den…

For each $u$ with $1\leq u\leq\lfloor(q+1)/2\rfloor$, let ${\mathcal C}_u$ be the classical MDS cyclic code of length $q+1$ over ${\mathrm{GF}}(q)$, and let…

Let $cmathcal{C}_{D_h}$ be the code considered above, and let $p=2$. Consider the supports of its minimum-weight codewords. Minimum-weight support 3-design conjecture. The supports…

Let $\mathcal{C}_{D_h}$ be the code defined in Theorem 1 of the source, and consider its minimum weight codewords. 3-design conjecture. The minimum weight codewords of…

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…

Affine-map conjecture. All equivalent cyclic codes are equivalent through an affine map of this form on cyclotomic cosets.

A projective irreducible cyclic code is an irreducible cyclic code satisfying the projectivity condition, and a two-weight code is a code with exactly two distinct nonzero Hamming…

Lubotzky's intermediate uncertainty principle conjecture. Although this strongest inequality can fail in particular settings over finite fields, intermediate variants—inequalities…

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

Let $\Gamma(n,k,2)$ denote the largest number of nonzero weights in a binary cyclic code of length $n$ and dimension $k$. Fix $k$ of the form … For each $i$, let $h_i(x)$ be an irr…

Let $\mathcal{C}$ denote any of the reversible cyclic codes constructed in the paper, and let the BCH bound provide a lower bound on its minimum distance. Exact BCH-bound conjectur…

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 $\mathcal{E}_{m,t,2^h}$ be the cyclic code based on Euclidean geometry considered in the paper. Its minimum nonzero codeword weight is $(2^{hm}+2)2^{m-t-1}-1$. Minimum-weight c…

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…