A class of index coding problems with rate 1/3

An index coding problem with $n$ messages has symmetric rate $R$ if all $n$ messages can be conveyed at rate $R$. In a recent work, a class of index coding problems for which symmetric rate $\frac{1}{3}$ is achievable was characterised using special properties of the side-information available at the receivers. In this paper, we show a larger class of index coding problems (which includes the previous class of problems) for which symmetric rate $\frac{1}{3}$ is achievable. In the process, we also obtain a stricter necessary condition for rate $\frac{1}{3}$ feasibility than what is known in literature.Comment: Shorter version submitted to ISIT 201

Index Coding: Rank-Invariant Extensions

An index coding (IC) problem consisting of a server and multiple receivers with different side-information and demand sets can be equivalently represented using a fitting matrix. A scalar linear index code to a given IC problem is a matrix representing the transmitted linear combinations of the message symbols. The length of an index code is then the number of transmissions (or equivalently, the number of rows in the index code). An IC problem ${\cal I}_{ext}$ is called an extension of another IC problem ${\cal I}$ if the fitting matrix of ${\cal I}$ is a submatrix of the fitting matrix of ${\cal I}_{ext}$. We first present a straightforward $m$\textit{-order} extension ${\cal I}_{ext}$ of an IC problem ${\cal I}$ for which an index code is obtained by concatenating $m$ copies of an index code of ${\cal I}$. The length of the codes is the same for both ${\cal I}$ and ${\cal I}_{ext}$, and if the index code for ${\cal I}$ has optimal length then so does the extended code for ${\cal I}_{ext}$. More generally, an extended IC problem of ${\cal I}$ having the same optimal length as ${\cal I}$ is said to be a \textit{rank-invariant} extension of ${\cal I}$. We then focus on $2$-order rank-invariant extensions of ${\cal I}$, and present constructions of such extensions based on involutory permutation matrices

Optimal Index Codes via a Duality between Index Coding and Network Coding

In Index Coding, the goal is to use a broadcast channel as efficiently as possible to communicate information from a source to multiple receivers which can possess some of the information symbols at the source as side-information. In this work, we present a duality relationship between index coding (IC) and multiple-unicast network coding (NC). It is known that the IC problem can be represented using a side-information graph $G$ (with number of vertices $n$ equal to the number of source symbols). The size of the maximum acyclic induced subgraph, denoted by $MAIS$ is a lower bound on the \textit{broadcast rate}. For IC problems with $MAIS=n-1$ and $MAIS=n-2$, prior work has shown that binary (over ${\mathbb F}_2$) linear index codes achieve the $MAIS$ lower bound for the broadcast rate and thus are optimal. In this work, we use the the duality relationship between NC and IC to show that for a class of IC problems with $MAIS=n-3$, binary linear index codes achieve the $MAIS$ lower bound on the broadcast rate. In contrast, it is known that there exists IC problems with $MAIS=n-3$ and optimal broadcast rate strictly greater than $MAIS$

A Survey on Fixed Divisors

In this article, we compile the work done by various mathematicians on the topic of the fixed divisor of a polynomial. This article explains most of the results concisely and is intended to be an exhaustive survey. We present the results on fixed divisors in various algebraic settings as well as the applications of fixed divisors to various algebraic and number theoretic problems. The work is presented in an orderly fashion so as to start from the simplest case of $\Z,$ progressively leading up to the case of Dedekind domains. We also ask a few open questions according to their context, which may give impetus to the reader to work further in this direction. We describe various bounds for fixed divisors as well as the connection of fixed divisors with different notions in the ring of integer-valued polynomials. Finally, we suggest how the generalization of the ring of integer-valued polynomials in the case of the ring of $n \times n$ matrices over $\Z$ (or Dedekind domain) could lead to the generalization of fixed divisors in that setting.Comment: Accepted for publication in Confluentes Mathematic

Coded Caching via Line Graphs of Bipartite Graphs

We present a coded caching framework using line graphs of bipartite graphs. A clique cover of the line graph describes the uncached subfiles at users. A clique cover of the complement of the square of the line graph gives a transmission scheme that satisfies user demands. We then define a specific class of such caching line graphs, for which the subpacketization, rate, and uncached fraction of the coded caching problem can be captured via its graph theoretic parameters. We present a construction of such caching line graphs using projective geometry. The presented scheme has a rate bounded from above by a constant with subpacketization level $q^{O((log_qK)^2)}$ and uncached fraction $\Theta(\frac{1}{\sqrt{K}})$, where $K$ is the number of users and $q$ is a prime power. We also present a subpacketization-dependent lower bound on the rate of coded caching schemes for a given broadcast setup.Comment: Keywords: coded caching based on projective geometry over finite field

Coded Caching based on Combinatorial Designs

We consider the standard broadcast setup with a single server broadcasting information to a number of clients, each of which contains local storage (called \textit{cache}) of some size, which can store some parts of the available files at the server. The centralized coded caching framework, consists of a caching phase and a delivery phase, both of which are carefully designed in order to use the cache and the channel together optimally. In prior literature, various combinatorial structures have been used to construct coded caching schemes. In this work, we propose a binary matrix model to construct the coded caching scheme. The ones in such a \textit{caching matrix} indicate uncached subfiles at the users. Identity submatrices of the caching matrix represent transmissions in the delivery phase. Using this model, we then propose several novel constructions for coded caching based on the various types of combinatorial designs. While most of the schemes constructed in this work (based on existing designs) have a high cache requirement (uncached fraction being $\Theta(\frac{1}{\sqrt{K}})$ or $\Theta(\frac{1}{K})$, $K$ being the number of users), they provide a rate that is either constant or decreasing ($O(\frac{1}{K})$) with increasing $K$, and moreover require competitively small levels of subpacketization (being $O(K^i), 1\leq i\leq 3$), which is an extremely important parameter in practical applications of coded caching. We mark this work as another attempt to exploit the well-developed theory of combinatorial designs for the problem of constructing caching schemes, utilizing the binary caching model we develop.Comment: 10 pages, Appeared in Proceedings of IEEE ISIT 201