8 problems

Let $n$, $k$, $d$, and $t$ be integers such that $n-1\geq d\geq k\geq 2$ and $d\leq t(d-k+1)$. Let … The main theorem establishes that, when $\alpha\leq 3003$, there exists an…

For a distributed storage system with parameters $(n,k,d)$, cascade codes are defined as a family of codes constructed from determinant codes. Cascade-code conjecture. Cascade code…

For a distributed storage system with parameters $(n,k,d)$, where $n$ is the total number of nodes, $k$ is the number of nodes sufficient for data recovery, and $d$ is the number o…

Tamo et al.'s conjecture. The longest systematic length $k$ is of order

Let ${\cal M}$ be the file size, and let $b$, $\rho$, $sigma$, $k$, $d$, and $k_c$ be the block-failure resilient coding parameters. Assume $d_r\geq k_c$ and $sigma>rho$. BFR-MSR a…

Exponential subpacketization conjecture. Attaining this bandwidth requires $t$ to be exponentially large in $k$.

Consider an MDS array code with disk capacity $l$, $k$ systematic nodes, and $r$ parity nodes, where $r$ is fixed independently of $l$. Tamo–Wang–Barg conjecture. For a given disk…

An $(n,k)$ MDS array code stores data across $n$ disks, with $k$ systematic nodes and a fixed number of parity nodes. Let $l$ denote the sub-packetization factor, or vector length…