19 problems

Let $p>5$ be an odd prime, let $C_{NQ}\subset {\mathbb F}^{2p}$ be the quasi-quadratic residue code, and let $C'$ be the code spanned by $C_{NQ}$ and the all-ones codeword. Let…

Let $C$ be a formally self-dual binary code, and let $Z_C(T)$, or equivalently $\zeta_C(s)=Z_C(2^{-s})$, denote its Duursma zeta function. For a self-dual code, call $C$ extremal w…

Let $q$ be a prime power, and let $n$ denote the length of a Hermitian self-dual generalized Reed–Solomon (GRS) code. Guo–Li's nonexistence conjecture. There are no Hermitian self-…

Let $q$ be a prime power and let $n$ be an even length satisfying … A Hermitian self-dual MDS generalized Reed–Solomon (GRS) code is one that is self-dual under the Hermitian inner…

Let $C$ be a binary linear self-dual $[72,36,16]$ code, and let $\sigma$ be a permutation in its permutation automorphism group $\operatorname{PAut}(C)$. Automorphism-group conject…

Let $C$ be a putative extremal binary self-dual $[72,36,16]$ code, and let $C_3$ denote the cyclic group of order $3$. Automorphism-group conjecture. The permutation automorphism g…

Let $C$ be a polycyclic code over $\mathbb{F}_q$ with generator $g$. Self-dual polycyclic-code structure conjecture. The code $C$ is self-dual if and only if its generator has the…

Let $C$ be a polycyclic code over the finite field $\bbF_q$, with generator polynomial $g$. Self-dual classification conjecture. The code $C$ is self-dual if and only if its genera…

Conjecture. The displayed formula for $|\mathcal{W}_0^{(s)}|$, and hence the resulting formula for $\omega_\lambda$, should hold for the stated range of parameters. The claim has b…

Uniqueness conjecture. There are no other $[54,27,10]$ self-dual cubic codes over ${\mathbb F}_2$.

Completeness conjecture. There exist no more binary cubic self-dual codes with length $54$, $60$, and $66$.

Let $p=2$, let $k$ and $t$ be parameters with $0\le t\le 2^{k-1}-1$, and let $\alpha\in\mathbb F_{2^m}$ be nonzero. Consider polynomials $g(x)\in\mathbb F_{2^m}[x]$ satisfying … wi…

Let $QR_{20}$ be the self-dual $[20,10,9]$ code over the finite field $\operatorname{GF}(7)$. Two codes are equivalent if one can be obtained from the other by the standard coordin…

Monroe's conjecture. For any distance, lexicographic greedy generation of self-orthogonal codes produces a self-dual code.

Let $N_{8,I}(2k)$ and $N_{8,II}(2k)$ denote, respectively, the numbers of inequivalent Type I and Type II $\mathbb{Z}_{2k}$-codes of length $8$. The length-8 code-count conjecture.…

Let $q_{48}$ be the length-$48$ Type II code considered above, and let $\operatorname{ODP}^{dic}[q_{48}]$ denote its optimum distance profile in dictionary order. A subcode chain f…

Let $SD_{20}$ denote the number of inequivalent binary self-dual codes of length $40$. Upper-bound conjecture. The number of the inequivalent binary self-dual codes of length $40$…

Let $C$ be a self-dual ternary code of length $24$ containing the all-one vector. The twisted construction associates to $C$ the unimodular neighbor … where … and…

A self-dual $[24,12,6]$ code is a ternary self-dual linear code of length $24$, dimension $12$, and minimum weight $6$; its automorphism group consists of the coordinate permutatio…