On the Broadcast Rate of Index Coding Problems with Symmetric and Consecutive Interference

A single unicast index coding problem (SUICP) with symmetric and consecutive interference (SCI) has $K$ messages and $K$ receivers, the $k$th receiver $R_k$ wanting the $k$th message $x_k$ and having interference $\mathcal{I}_k= \{x_{k-U-m},\dots,x_{k-m-2},x_{k-m-1}\}\cup\{x_{k+m+1}, x_{k+m+2},\dots,x_{k+m+D}\}$ and side-information $\mathcal{K}_k=(\mathcal{I}_k \cup x_k)^c$. In this paper, we derive a lowerbound on the broadcast rate of single unicast index coding problem with symmetric and consecutive interference (SUICP(SCI)). In the SUICP(SCI), if $m=0$, we refer this as single unicast index coding problem with symmetric and neighboring interference (SUICP(SNI)). In our previous work\cite{VaR5}, we gave the construction of near-optimal vector linear index codes for SUICP(SNI) with arbitrary $K,D,U$. In this paper, we convert the SUICP(SCI) into SUICP(SNI) and give the construction of near-optimal vector linear index codes for SUICP(SCI) with arbitrary $K,U,D$ and $m$. The constructed codes are independent of field size. The near-optimal vector linear index codes of SUICP(SNI) is a special case of near-optimal vector linear index codes constructed in this paper for SUICP(SCI) with $m=0$. In our previous work\cite{VaR6}, we derived an upperbound on broadcast rate of SUICP(SNI). In this paper, we give an upperbound on the broadcast rate of SUICP(SCI) by using our earlier result on the upperbound on the broadcast rate of SUICP(SNI). We derive the capacity of SUICP(SCI) for some special cases.