Combinatorial Limitations of Average-Radius List Decoding

We study certain combinatorial aspects of list-decoding, motivated by the exponential gap between the known upper bound (of O(1/γ)) and lower bound (of Ω p (log(1/γ))) for the list-size needed to list decode up to error fraction p with rate γ away from capacity, i.e., 1 − h(p) − γ (here \(p\in (0, \frac{1}{2})\) and γ > 0). Our main result is the following:

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