List-Decoding Capacity Implies Capacity on the q-ary Symmetric Channel

It is known that the Shannon capacity of the q-ary symmetric channel (qSC) is the same as the list-decoding capacity of an adversarial channel, raising the question of whether there is a formal (and black-box) connection between the two. We show that there is: Any linear code C ⊆ F n q that has minimum distance d min = ω q 3 and achieves list-decoding capacity also achieves capacity on the qSC.