22 problems

Let $C$ be a doubly even binary code of length $n$, and let $\Lambda_C$ be the inverse image of $C$ under the natural map … Let $\mathbb{Z}[\frac{1}{2}\Lambda_C]$ denote the ring g…

Let $C\subset \mathbb{F}_2^n$ be a maximal doubly even binary code, and let $I=(i_0,i_1,\ldots,i_{n-1})$. Consider the Clifford algebra $\mathbb{C}_n$ and the rational Clifford alg…

Choi et al.'s conjecture. There exists a unique binary optimal $[n,k,d_L(n,k)]$ code $C$ which is even-like and contains ${\bf1}$ if and only if

Let $A_2(r,s)$ denote the maximum size of a binary code of block length $r$ and minimum Hamming distance $s$. For $j=o\left(r^{1/3}\right)$, consider binary codes with distance…

Let $d(n,k)$ denote the largest minimum distance of a binary linear $[n,k]$ code, and let $d_{so}(n,k)$ denote the largest minimum distance of a binary self-orthogonal $[n,k]$ code…

Let $C$ be a binary quasi group code of length $n=2m>2$, and let $\operatorname{PAut}(C)$ denote its permutation automorphism group. Nonexistence conjecture. There is no binary qua…

Optimal binary linear insertion-deletion code conjecture. If $n>2k$, $\mathbf{1}\notin\mathcal C$, and $\mathcal C$ is optimal with respect to the strict half-Singleton bound, then…

Gilbert–Varshamov optimality conjecture. There does not exist any binary code exceeding the Gilbert–Varshamov lower bound.

A polycyclic code over the binary field $\mathbb{F}_2$ may be self-dual, self-orthogonal, or dual-containing. Binary nonexistence conjecture. There is no self-dual, self-orthogonal…

Let $C$ be a polycyclic code over the binary field $\bbF_2$. Binary nonexistence conjecture. There are no polycyclic codes over $\bbF_2$ that are self-dual, self-orthogonal, or dua…

Parameter and weight-distribution conjecture. The parameters of $\mathcal{C}_f$ should be the same as those of the code in the cited theorem, namely the theorem's parameters for th…

The weight-distribution conjecture. The polynomial $f(x)$ is two-to-one over ${\mathbb F}_{2^n}$, the code $\mathcal{C}_{D(f)}$ has parameters

Let $r$, $s$, and $L$ be nonnegative integers with $L\geqslant 2s+r+1$, and let $C\subseteq {\cal W}_s$ be a minimal $(L,s,r)$ PPRIC code. An MIPPR word is a word associated with…

Completeness conjecture. There are no doubly even binary $[45,8,20]$ codes beyond those obtained in the cited classification.

A binary code is asymptotically optimal in the sense defined in Theorem 7 if its performance achieves the capacity of binary erasure channels (BECs) with the optimal scaling expone…

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 $\Lambda$ be a lattice in dimension $n\in\{10,11,12\}$, and let $\Lambda'$ denote the sublattice generated by the minimal vectors. The quotient $\Lambda/\Lambda'$ is 2-elementa…

Let $M(n,k)$ denote the maximum number of minimal codewords among binary linear codes of parameters $[n,k]$. Let the lower bounds listed in Table 2 of the paper be the bounds obtai…

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 $40$…

Keith's conjecture.

Let $q$ be an odd prime power, and let ${\mathbf B}$ and ${\mathbf B}_0$ be the incidence matrices of internal points versus secant lines and external points versus passant lines,…

Let $Q^m$ be the $m$-dimensional binary Hamming space, let $M\subseteq Q^m$ be a double-MDS-code, and let $V(Q^m)\setminus M$ denote its complement. Potapov's conjecture. The code…