20 problems

Alrabiah–Guruswami conjecture. This value of $N$ is exactly tight for high-rate MSR codes.

Let $\mathcal{C}$ be an $[n,k]$ code, and let $d_s^{\mathcal{C}}$ denote the quantity appearing in the necessary condition … An MDS-PIR capacity-achieving matrix…

An allocation distributes storage across $n$ nodes, and its recovery delay is denoted by $D$; the parameters are the node count $n$, request rate $\lambda$, and threshold $T$. An a…

The bandwidth trade-off curve describes the achievable operating points for cooperative node repair in distributed storage systems with base stations and link constraints. Closed-f…

In the fixed-size access model, let $N$ be the number of nodes, $m$ the redundancy level, $r$ the number of accessed nodes, and $b1$ the spreading parameter, with minimal spreading…

Continuous standard repair optimality conjecture. The factor

Divisibility conjecture. The condition $k\mid m$ is necessary for vertical MDS array codes to be MUB codes.

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…

Let $m$ be the coded file size ratio, and let $\alpha$ be the allocation parameter maximizing the average service rate $\mu_s(\alpha)$. The claim applies to both the fixed-size acc…

Let $p$ be the access failure probability for each node, let $\alpha$ be the allocation parameter, and let $\mu_s(\alpha)$ denote the average service rate under probabilistic acces…

Let $r$ denote the number of accessed nodes, let $\alpha$ be the allocation parameter, and let $\mu_s(\alpha)$ denote the average service rate under fixed-size access. Fixed-size a…

Consider shared-memory emulation algorithms with a bounded number of phases, at most one of which sends a value-dependent message of size $\Theta(|\mathcal{V}|)$; the other value-d…

Theorem 4 concerns shared-memory emulation algorithms whose value-dependent messages satisfy its stated assumptions. Some algorithms instead send value-dependent messages in two ph…

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…

Consider a block-failure resilient storage system with $b$ blocks, where each block contains $n/b$ nodes, and let $\rho$ and $\sigma$ denote the relevant numbers of blocks unavaila…

Let $R(x)$ be the reliability of a constituent component, let $\lambda(x)$ be its failure rate, and let $\mu_c(x,n,r)$ denote the hazard rate per data component of a horizontal sys…

Let $n$ denote the code parameter, let $V$ denote the matrix computed by the proposed repair-by-transfer algorithm, and let the comparison code of Shah be the repair-by-transfer co…

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…