On the Minrank of Symmetric and Neighboring Side-information Index Coding Problems

The length of an optimal scalar linear index code of a single unicast index coding problem (SUICP) is equal to the minrank of its side-information graph. A single unicast index coding problem is called symmetric neighboring and consecutive (SNC) side-information problem if it has $K$ messages and $K$ receivers, the $k\mathrm {t}\mathrm {h}$ receiver $R_{k}$ wanting the $k\mathrm {t}\mathrm {h}$ 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 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 analyse the properties of minrank of SUICP(SNC) side-information graph. We derive the MAIS (${G}$) of side-information graph $G$ of SUICP(SNC). For arbitrary $K, D$ and $U$, we construct scalar linear index codes for SUICP(SNC) with length $\displaystyle \lceil \frac {K}{U+1}\rceil - \displaystyle \lfloor \frac {D-U}{U+1}\rfloor $. We obtain the minrank of SUICP(SNC) side-information graph and show that the length of the constructed scalar linear index codes is equal to minrank of SUICP(SNC) side-information graph for some combinations of K, D and ${U}$.