Optimal Index Code With Rate 4 Over F₂

High throughput broadcasting systems requires efficient broadcasting that can broadcast the information source symbols with optimal rate (i.e., maximizing the usage of broadcast channel which exploits the broadcast nature of the wireless medium through index coding). Index coding is the elegant and beautiful idea, which is used to transform the information source symbols for efficient broadcasting. Efficient broadcasting means minimizing the number of transmissions, which facilitate to achieve high throughput in broadcasting systems is the main goal for index coding. In index coding, determining the optimal rate of an index code for the class of index coding problem is the central open problem. In this paper, we present class <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb{I V}$ </tex-math></inline-formula> index coding problems, a new class of index coding problems with the size of maximum acyclic induced subgraph is 4. The main goal of this work is to characterize the optimal rate of class <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb{I V}$ </tex-math></inline-formula> index coding problems over finite field with 2 elements <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\left(\mathbb{F}_2\right)$ </tex-math></inline-formula>. We present that optimal linear index code of the class <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb{I V}$ </tex-math></inline-formula> index coding problem over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb{F}_2$ </tex-math></inline-formula> achieves the rate 4 on the broadcast rate. To present this new result, results of index coding network coding duality are used. We also present that there exists index coding problems with size of maximum acyclic induced subgraph is 4 and optimal broadcast rate greater than 4. We use the technique interference alignment using alignment graph and the conflict hyper graph to present this contrast.