26 problems

Small-field conjecture. A finite field of size

Existence conjecture. Maximum distance profile codes exist for every transmission rate.

Existence conjecture. For all $n>k>0$ and for all $\delta\geq 0$ there exists an $(n,k,\delta)$ code over a sufficiently large field which is both strongly MDS and has a maximum di…

Superregular matrix existence conjecture. For every integer $b3\geq 5$, there exists a $b3\times b3$ superregular matrix over

MacWilliams identity conjecture. The matrix

Let $\mathcal C$ and $\mathcal C'$ be codes in $\mathbb F(\!(z)\!)^n$, and let $\bar{\Lambda}(\mathcal C)$ denote the generalized adjacency matrix of $\mathcal C$. Dual invariance…

Let $\mathcal C$ and $\mathcal C'$ be one-dimensional convolutional codes over an arbitrary field. General-field conjecture. If $\mathcal C$ and $\mathcal C'$ have the same general…

Let $\mathcal C$ and $\mathcal C'$ be convolutional codes with generalized adjacency matrices $\bar{\Lambda}(\mathcal C)$ and $\bar{\Lambda}(\mathcal C')$. MacWilliams duality conj…

Kuijper–Pinto's conjecture. Every convolutional code over $\mathbb{Z}_{p^r}$ admits a minimal $p$-encoder.

Smaller-extension-degree conjecture. The construction is MDP also for smaller values of $d$. The conjecture would reduce the finite-field extension degree needed by this matrix-com…

Let a code use an $m$-bit CRC and let its decoder have a maximum list size. CRC list-size conjecture. A maximum list size of around $2^m$ is necessary for the code to achieve perfo…

Superregular Toeplitz matrix conjecture. For every $r\geq 5$, there exists a superregular lower triangular Toeplitz matrix of order $r$ over $\mathbb F_{2^{r-2}}$.

Let SBCCs denote sparsely braided convolutional codes, including bitwise and blockwise constructions, and let their overall constraint length be the relevant complexity parameter.…

Let $m$-MSR convolutional codes be constructed using superregular matrices of the type proposed in Almeida, Napp and Pinto (2013), Almeida and coauthors (2016), and Martínez-Peñas…

Non-equivalence conjecture. Not every LRCC attaining this bound for some $j\in\mathbb{N}$ is a partial $j$-MDS convolutional code.

Consider a minimal convolutional code with $m$ memory elements and a degree-$v$ distance-spectrum-optimal CRC code, used together under serial list Viterbi decoding at a fixed SNR.…

Let $m$ be the degree of the CRC code, and let $\operatorname{P}^{1}_{\operatorname{UE}}$ and $\operatorname{P}^{1}_{\operatorname{F}}$ denote, respectively, the undetected-error p…

Let $cgamma$ be an integer with $cgamma dagger 5$. A matrix is superregular Toeplitz if it is both superregular and Toeplitz. Let $n$, $k$, and $cdelta$ be convolutional-code p…

Convolutional sum-rank-metric codes are convolutional codes in which each block is measured using an $\ell$-shot sum-rank metric, rather than the rank metric. The comparison is wit…

Let $\boldsymbol{u}^k\stackrel{\triangle}{=}u_1u_2\cdots u_k$ be the transmitted information sequence. In the generalized Viterbi algorithm, let $\nu$ be the encoder constraint len…

Kumar's conjecture. Every convolutional code over $\mathbb Z_{p^r}$ admits a noncatastrophic $p$-encoder. Noncatastrophic $p$-encoders provide a useful encoder-level finiteness pro…

Let $l\geq 5$. A superregular $l\times l$ Toeplitz matrix is a matrix with the superregularity property used in the construction of MDP convolutional codes. Gluesing-Luerssen's con…

Free-distance conjecture. It is conjectured that

Let ${\mathbb Z}_{p^r}$ be the finite ring of integers modulo $p^r$, and let a convolutional code over ${\mathbb Z}_{p^r}$ be encoded by a $p$-encoder. A code is catastrophic if it…

Let $(n,k,\delta)$ be convolutional-code parameters. A convolutional code has the MDP property when its column distances grow at the maximum possible rate for the given parameters,…