From Bit to Block: Decoding on Erasure Channels

We provide a general framework for bounding the block error threshold of a linear code $C \subseteq \mathbb{F}_{2}^{N}$ over the erasure channel in terms of its bit error threshold. Our approach relies on understanding the minimum support weight of any $r$-dimensional subcode of $C$, for all small values of $r$. As a proof of concept, we use our machinery to obtain a new proof of the celebrated result that Reed-Muller codes achieve capacity on the erasure channel with respect to block error probability.