Task-Based Solutions to Embedded Index Coding

In the <italic>index coding</italic> problem a sender holds a message <inline-formula> <tex-math notation="LaTeX">$x \in \{0,1\}^{n}$ </tex-math></inline-formula> and wishes to broadcast information to <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> receivers in a way that enables the <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula>th receiver to retrieve the <inline-formula> <tex-math notation="LaTeX">$i$ </tex-math></inline-formula>th bit <inline-formula> <tex-math notation="LaTeX">$x_{i}$ </tex-math></inline-formula>. Every receiver has prior side information comprising a subset of the bits of <inline-formula> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula>, and the goal is to minimize the length of the information sent via the broadcast channel. Porter and Wootters have recently introduced the model of <italic>embedded index coding</italic>, where the receivers also play the role of the sender and the goal is to minimize the total length of their broadcast information. An embedded index code is said to be <italic>task-based</italic> if every receiver retrieves its bit based only on the information provided by one of the receivers. This paper studies the effect of the task-based restriction on linear embedded index coding. It is shown that for certain side information maps there exists a linear embedded index code of length quadratically smaller than that of any task-based embedded index code. The result attains, up to a multiplicative constant, the largest possible gap between the two quantities. The proof is by an explicit construction and the analysis involves spectral techniques.

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