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

A single unicast index coding problem (SUICP) with symmetric neighboring and consecutive side-information (SNCS) has $K$ messages and $K$ receivers, the $k$th receiver $R_k$ wanting the $k$th message $x_k$ and having the side-information $\mathcal{K}_k=\{x_{k-U},\dots,x_{k-2},x_{k-1}\}\cup\{x_{k+1}, x_{k+2},\dots,x_{k+D}\}$. The single unicast index coding problem with symmetric neighboring and consecutive side-information, SUICP(SNCS), is motivated by topological interference management problems in wireless communication networks. Maleki, Cadambe and Jafar obtained the symmetric capacity of this SUICP(SNCS) and proposed optimal length codes by using Vandermonde matrices. In our earlier work, we gave optimal length $(U+1)$-dimensional vector linear index codes for SUICP(SNCS) satisfying some conditions on $K,D$ and $U$ \cite{VaR1}. In this paper, for SUICP(SNCS) with arbitrary $K,D$ and $U$, we construct optimal length $\frac{U+1}{\text{gcd}(K,D-U,U+1)}$-dimensional vector linear index codes. We prove 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 proposed construction gives optimal length scalar linear index codes for the SUICP(SNCS) if $(U+1)$ divides both $K$ and $D-U$. The proposed construction is independent of field size and works over every field. We give a low-complexity decoding for the SUICP(SNCS). By using the proposed decoding method, every receiver is able to decode its wanted message symbol by simply adding some index code symbols (broadcast symbols).Comment: 13 pages, 1 figure and 5 table

On Approximating the Sum-Rate for Multiple-Unicasts

We study upper bounds on the sum-rate of multiple-unicasts. We approximate the Generalized Network Sharing Bound (GNS cut) of the multiple-unicasts network coding problem with $k$ independent sources. Our approximation algorithm runs in polynomial time and yields an upper bound on the joint source entropy rate, which is within an $O(\log^2 k)$ factor from the GNS cut. It further yields a vector-linear network code that achieves joint source entropy rate within an $O(\log^2 k)$ factor from the GNS cut, but \emph{not} with independent sources: the code induces a correlation pattern among the sources. Our second contribution is establishing a separation result for vector-linear network codes: for any given field $\mathbb{F}$ there exist networks for which the optimum sum-rate supported by vector-linear codes over $\mathbb{F}$ for independent sources can be multiplicatively separated by a factor of $k^{1-\delta}$, for any constant ${\delta>0}$, from the optimum joint entropy rate supported by a code that allows correlation between sources. Finally, we establish a similar separation result for the asymmetric optimum vector-linear sum-rates achieved over two distinct fields $\mathbb{F}_{p}$ and $\mathbb{F}_{q}$ for independent sources, revealing that the choice of field can heavily impact the performance of a linear network code.Comment: 10 pages; Shorter version appeared at ISIT (International Symposium on Information Theory) 2015; some typos correcte

On Critical Index Coding Problems

The question of under what condition some side information for index coding can be removed without affecting the capacity region is studied, which was originally posed by Tahmasbi, Shahrasbi, and Gohari. To answer this question, the notion of unicycle for the side information graph is introduced and it is shown that any edge that belongs to a unicycle is critical, namely, it cannot be removed without reducing the capacity region. Although this sufficient condition for criticality is not necessary in general, a partial converse is established, which elucidates the connection between the notion of unicycle and the maximal acylic induced subgraph outer bound on the capacity region by Bar-Yossef, Birk, Jayram, and Kol.Comment: 5 pages, accepted to 2015 IEEE Information Theory Workshop (ITW), Jeju Island, Kore