Low Complexity Decoding and Capacity of Index Coding Problems with Symmetric Side-Information

A single unicast index coding problem (SUICP) with symmetric side-information has K messages and K receivers, the $kth$ receiver $R_{k}$ wants $x_{k}, R_{k}$ has some subset of messages as side-information and the side-information is symmetric to its wanted message $x_{k}$. Maleki, Cadambe and Jafar studied various symmetric index coding problems because of their importance in topological interference management problems. In our previous work, we constructed binary matrices of size $m\times n$ for any given arbitrary positive integers m and n such that any n adjacent rows of this matrix are linearly independent. We refer these matrices as Adjacent Independent Row (AIR) matrices. We designed optimal and near-optimal vector linear index codes for various symmetric SUICPs by using AIR matrices. To design the optimal and near-optimal vector linear index codes, we convert the respective symmetric SUICP into an SUICP with symmetric neighboring and consecutive (SNC) side-information. Then, we use AIR matrix to encode the SUICP with SNC side-information. Hence, low-complexity decoding of SUICP with SNC side-information is important for efficient decoding of optimal and near-optimal index codes for various symmetric SUICPs. We analyse some of the combinatorial properties of AIR matrices in this work. By using these properties, we provide a low-complexity decoding for SUICP with SNC side-information. The low-complexity decoding explicitly identifies the set of broadcast symbols required at every receiver to decode its wanted message. By using low-complexity decoding, we find the capacity of SUICP with symmetric side-information $\mathcal{K}_{k} = \{x_{k+g},\ x_{k+2g}, \ldots , x_{k+tg}\}$, where $g=gcd(K,\ D)$ and $t= \frac{D}{g}$ for any positive integer $D\lt K.$