28 problems

Binary scalar linear optimality conjecture. Binary scalar linear index codes are optimal, meaning that

Secure coded caching involves a server storing messages and users with local caches who request messages over a broadcast link, while secure index coding involves an adversary that…

Let $\mathcal{I}=\{(i\mid A_i),i\in[m]\}$ be an index coding instance. The ICC and fractional-partitioned clique-cover coding schemes have broadcast rates…

Consider the consecutive and symmetric embedded index coding problems studied in the paper, and let the normalized rate denote the number of transmitted bits normalized by the tota…

Sub-packetization rate conjecture. The normalized rate achieved by the sub-packetization scheme is lower than $\frac{N-s+1}{N}$, the normalized rate achieved using a scalar linear…

Let SUICP(SNC) denote the single-unicast index-coding problem with symmetric neighboring side information, with $K$ receivers and messages, interference parameter $D$, and side-inf…

Consider the six sub-cases of Case I with fully-participated interactions treated in this paper, and let $t$-bit messages be used for any finite $t$. The paper gives achievable bro…

Let $\mathcal{D}^k$, $k \in \{13,14,\cdots,25\}$, be a side-information digraph whose sub-digraphs $\mathcal{D}_{i}^{k,\mathcal{P}}$, $i \in \{1,2,3\}$, have any type of interactio…

Let $m$ be the number of messages and let $t$ be a positive integer. For integers $s_{\min}$ and $s_{\max}$ satisfying … consider the complete--$[s_{\min}:s_{\max}]$ PICOD$(t)$ pro…

ILC converse conjecture. Whenever the ILC condition is violated, the digraph is not an IC structure.

Minimal dimensionality conjecture. For every SUICP(SCNC) not satisfying

A non-symmetric index code is a code whose side-information structure is not symmetric in the sense considered in Corollary. Criticality conjecture. These non-symmetric codes are c…

Let $G$ be a directed graph on $n$ vertices, and let $G^i$ denote its shift by $i$ modulo $n$. Write $\mathsf{minrk}_2(G^i)$ for the minimum length of a linear index-coding scheme…

Let $G=(V,A)$ be a directed graph on $n$ vertices with maximum out-degree at most $n^{\varepsilon}$. For each $i\in\{0,\ldots,n-1\}$, let $G^i$ be the graph obtained by shifting ev…

Let $x\in\{0,1\}^n$ and, for each $i\in\{0,\ldots,n-1\}$, consider the $i$th cyclic shift of $x$. A pre-wired circuit has each output depending on at most $n^{\varepsilon}$ hard-wi…

Let a unicast index coding problem have independent messages, and let its side-information graph encode which messages are known to which receivers. For a field $\mathbb{F}$, write…

Let $\mathcal G$ be a side-information graph, and let $\beta(\mathcal G)$ denote the minimum broadcast rate over all index codes, including nonlinear codes. Let…

Let $D'$ be a minimal partial clique, meaning that its savings cannot be matched or exceeded by the sum of the savings of any partition into smaller subdigraphs. Let $\delta^+(D')$…

Let $D$ be a digraph. Write $\ell_{\mathsf{ICC}}(D)$ and $\ell_{\mathsf{PC}}(D)$ for the index-code lengths produced by the interlinked-cycle-cover and partial-clique-cover schemes…

Let $D_K$ be a $K$-$\mathsf{IC}$ structure, and let $t\geq 1$ be the number of bits in each message. Write $\ell_{\mathsf{ICC}}(D_K)$ for the scalar linear index-code length of the…

An index coding problem consists of messages, receivers, and side-information constraints; the paper organizes its messages into alignment sets, with type-2 alignment sets subject…

Conjecture on encoding at the threshold. At $c=ab$, any method of encoding $\vec{w}_2$ and $\vec{w}_3$ that satisfies the decodability condition at Users 2 and 3 also allows User 1…

Let $D_K$ be a $K$-$\mathsf{GIC}$ digraph, and let each message have $t\geq 1$ bits. Write $\ell_{\mathsf{GICC}}(D_K)$ for the scalar index-code length produced by the GICC scheme,…

Let a USCS graph be a graph whose edges all lie on directed cycles. Suppose that the graph supports a given set of rates for asymptotic non-linear index coding, and let $e$ be an e…

Let $G$ be a graph in the constrained setting of graph Index Coding. Denote by $\beta$ the vector broadcast rate and by $\beta_1$ the scalar broadcast rate. Ceiling conjecture. The…