Low-Complexity Decoding for Symmetric, Neighboring and Consecutive Side-information Index Coding Problems

The capacity of symmetric, neighboring and consecutive side-information single unicast index coding problems (SNC-SUICP) with number of messages equal to the number of receivers was given by Maleki, Cadambe and Jafar. For these index coding problems, an optimal index code construction by using Vandermonde matrices was proposed. This construction requires all the side-information at the receivers to decode their wanted messages and also requires large field size. In an earlier work, we constructed binary matrices of size $m \times n (m\geq n)$ such that any $n$ adjacent rows of the matrix are linearly independent over every field. Calling these matrices as Adjacent Independent Row (AIR) matrices using which we gave an optimal scalar linear index code for the one-sided SNC-SUICP for any given number of messages and one-sided side-information. By using Vandermonde matrices or AIR matrices, every receiver needs to solve $K-D$ equations with $K-D$ unknowns to obtain its wanted message, where $K$ is the number of messages and $D$ is the size of the side-information. In this paper, we analyze some of the combinatorial properties of the AIR matrices. By using these properties, we present a low-complexity decoding which helps to identify a reduced set of side-information for each users with which the decoding can be carried out. By this method every receiver is able to decode its wanted message symbol by simply adding some index code symbols (broadcast symbols). We explicitly give both the reduced set of side-information and the broadcast messages to be used by each receiver to decode its wanted message. For a given pair or receivers our decoding identifies which one will perform better than the other when the broadcast channel is noisy.Comment: Some minor changes made in Algorithm 1. 11 Figures, 3 Table

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.Comment: 6 pages, 1 figure, 2 tables. arXiv admin note: text overlap with arXiv:1801.0040

Groupcast Index Coding Problem: Joint Extensions

The groupcast index coding problem is the most general version of the classical index coding problem, where any receiver can demand messages that are also demanded by other receivers. Any groupcast index coding problem is described by its \emph{fitting matrix} which contains unknown entries along with $1$'s and $0$'s. The problem of finding an optimal scalar linear code is equivalent to completing this matrix with known entries such that the rank of the resulting matrix is minimized. Any row basis of such a completion gives an optimal \emph{scalar linear} code. An index coding problem is said to be a joint extension of a finite number of index coding problems, if the fitting matrices of these problems are disjoint submatrices of the fitting matrix of the jointly extended problem. In this paper, a class of joint extensions of any finite number of groupcast index coding problems is identified, where the relation between the fitting matrices of the sub-problems present in the fitting matrix of the jointly extended problem is defined by a base problem. A lower bound on the \emph{minrank} (optimal scalar linear codelength) of the jointly extended problem is given in terms of those of the sub-problems. This lower bound also has a dependence on the base problem and is operationally useful in finding lower bounds of the jointly extended problems when the minranks of all the sub-problems are known. We provide an algorithm to construct scalar linear codes (not optimal in general), for any groupcast problem belonging to the class of jointly extended problems identified in this paper. The algorithm uses scalar linear codes of all the sub-problems and the base problem. We also identify some subclasses, where the constructed codes are scalar linear optimal.Comment: 9 page