A New Upperbound on the Broadcast Rate of Index Coding Problems with Symmetric Neighboring Interference

Abstract

A single unicast index coding problem (SUICP) with symmetric neighboring interference (SNI) has equal number of $K$ messages and $K$ receivers, the $k$th receiver $R_{k}$ wanting the $k$th message $x_{k}$ and having the side-information $\mathcal{K}_{k}=(\mathcal{I}_{k} \cup x_{k})^c,$ where ${I}_k= \{x_{k-U},\dots,x_{k-2},x_{k-1}\}\cup\{x_{k+1}, x_{k+2},\dots,x_{k+D}\}$ is the interference with $D$ messages after and $U$ messages before its desired message. The single unicast index coding problem with symmetric neighboring interference (SUICP-SNI) is motivated by topological interference management problems in wireless communication networks. Maleki, Cadambe and Jafar obtained the capacity of this SUICP-SNI with $K$ tending to infinity and Blasiak, Kleinberg and Lubetzky for the special case of $(D=U=1)$ with $K$ being finite. Finding the capacity of the SUICP-SNI for arbitrary $K,D$ and $U$ is a challenging open problem. In our previous work, for an SUICP-SNI with arbitrary $K,D$ and $U$, we defined a set $\mathcal{\mathbf{S}}$ of $2$-tuples such that for every $(a,b)$ in that set $\mathcal{\mathbf{S}}$, the rate $D+1+\frac{a}{b}$ is achieved by using vector linear index codes over every finite field. In this paper, we give an algorithm to find the values of $a$ and $b$ such that $(a,b) \in \mathcal{\mathbf{S}}$ and $\frac{a}{b}$ is minimum. We present a new upperbound on the broadcast rate of SUICP-SNI and prove that this upper bound coincides with the existing results on the exact value of the capacity of SUICP-SNI in the respective settings.