From Bit to Block: Decoding on Erasure Channels

We provide a general framework for bounding the block error threshold of a linear code <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$C \subseteq \mathbb{F}_{2}^{N}$</tex> over the erasure channel in terms of its bit error threshold. Our approach relies on understanding the minimum support weight of any <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$r$</tex>-dimensional subcode of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$C$</tex>, for all small values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$r$</tex>. 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.