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A single unicast index coding problem (SUICP) with symmetric neighboring andconsecutive side-information (SNCS) has $K$ messages and $K$ receivers, the$k$th receiver $R_k$ wanting the $k$th message $x_k$ and having theside-information $\mathcal{K}_k=\{x_{k-U},\dots,x_{k-2},x_{k-1}\}\cup\{x_{k+1},x_{k+2},\dots,x_{k+D}\}$. The single unicast index coding problem withsymmetric neighboring and consecutive side-information, SUICP(SNCS), ismotivated by topological interference management problems in wirelesscommunication networks. Maleki, Cadambe and Jafar obtained the symmetriccapacity of this SUICP(SNCS) and proposed optimal length codes by usingVandermonde matrices. In our earlier work, we gave optimal length$(U+1)$-dimensional vector linear index codes for SUICP(SNCS) satisfying someconditions on $K,D$ and $U$ \cite{VaR1}. In this paper, for SUICP(SNCS) witharbitrary $K,D$ and $U$, we construct optimal length$\frac{U+1}{\text{gcd}(K,D-U,U+1)}$-dimensional vector linear index codes. Weprove that the constructed vector linear index code is of minimal dimension if$\text{gcd}(K-D+U,U+1)$ is equal to $\text{gcd}(K,D-U,U+1)$. The proposedconstruction gives optimal length scalar linear index codes for the SUICP(SNCS)if $(U+1)$ divides both $K$ and $D-U$. The proposed construction is independentof field size and works over every field. We give a low-complexity decoding forthe SUICP(SNCS). By using the proposed decoding method, every receiver is ableto decode its wanted message symbol by simply adding some index code symbols(broadcast symbols).

Reduced Dimensional Optimal Vector Linear Index Codes for Index Coding Problems with Symmetric Neighboring and Consecutive Side-information