Optimal Scalar Linear Index Codes for Symmetric and Neighboring Side-information Problems

A single unicast index coding problem (SUICP) is called symmetric neighboring and consecutive (SNC) side-information problem if it has $K$ messages and $K$ receivers, the $k$th receiver $R_{k}$ wanting the $k$th message $x_{k}$ and having the side-information $D$ messages immediately after $x_k$ and $U$ ($D\geq U$) messages immediately before $x_k$. Maleki, Cadambe and Jafar obtained the capacity of this SUICP(SNC) and proposed $(U+1)$-dimensional optimal length vector linear index codes by using Vandermonde matrices. However, for a $b$-dimensional vector linear index code, the transmitter needs to wait for $b$ realizations of each message and hence the latency introduced at the transmitter is proportional to $b$. For any given single unicast index coding problem (SUICP) with the side-information graph $G$, MAIS($G$) is used to give a lowerbound on the broadcast rate of the ICP. In this paper, we derive MAIS($G$) of SUICP(SNC) with side-information graph $G$. We construct scalar linear index codes for SUICP(SNC) with length $\left \lceil \frac{K}{U+1} \right \rceil - \left \lfloor \frac{D-U}{U+1} \right \rfloor$. We derive the minrank($G$) of SUICP(SNC) with side-information graph $G$ and show that the constructed scalar linear index codes are of optimal length for SUICP(SNC) with some combinations of $K,D$ and $U$. For SUICP(SNC) with arbitrary $K,D$ and $U$, we show that the length of constructed scalar linear index codes are atmost two index code symbols per message symbol more than the broadcast rate. The given results for SUICP(SNC) are of practical importance due to its relation with topological interference management problem in wireless communication networks.