Random linear binary codes have smaller list sizes than uniformly random binary codes

There has been a great deal of work establishing that random linear codes are as list-decodable as uniformly random codes, in the sense that a random linear binary code of rate $1 - H(p) - \epsilon$ is $(p,O(1/\epsilon))$-list-decodable. In this work, we show that in fact random linear binary codes are \em more \em list-decodable than uniformly random codes, in the sense that the constant in the $O(1/\epsilon)$ is strictly smaller for random linear codes than for uniformly random codes. For our upper bound on the list size of random linear codes, we strengthen an existential argument of (Guruswami, H{\aa}stad, Sudan and Zuckerman, IEEE Trans. IT, 2002) to hold with high probability. In addition to improving known list-size bounds, our argument works simultaneously for all values of $p$, while previous works obtaining $L = O(1/\epsilon)$ patched together different arguments to cover different parameter regimes. To complement our upper bound for random linear codes, we also improve an argument of (Guruswami, Narayanan, IEEE Trans. IT, 2014) to obtain an essentially tight lower bound on the list size of uniformly random codes. To demonstrate the applicability of these techniques, we use them to (a) obtain more information about the distribution of list sizes of random linear codes and (b) to prove a similar result for random linear rank-metric codes. More precisely, we prove that in some parameter regimes, random linear rank-metric codes have strictly smaller list sizes than uniformly random rank-metric codes; our upper bound improves upon a recent result of Guruswami and Resch, and we prove a new lower bound on the list size for uniformly rank metric codes.