Improved List-Decodability of Random Linear Binary Codes

There has been a great deal of work establishing that random linear codes are as list-decodable uniformly codes, in the sense binary code rate $1 - {H}(\text {p}) \epsilon $ is $( {p},\text {O}(1/\epsilon))$ -list-decodable with high probability. In this work, we show such {p}, {H}( {p})/\epsilon + 2)$ probability, for any {p} \in (0, 1/2)$ and $\epsilon > 0$ . addition to improving constant known list-size bounds, our argument—which quite simple—works simultaneously all values p, while previous works obtaining {L} = \text {O}(1/\epsilon)$ patched together different arguments cover parameter regimes. Our approach strengthen an existential argument (Guruswami, Hastad, Sudan Zuckerman, IEEE Trans. IT, 2002) hold To complement upper bound also improve Narayanan, 2014) obtain essentially tight lower $1/\epsilon on list size codes; implies fact more than sizes strictly smaller. demonstrate applicability these techniques, use them (a) information about distribution (b) prove similar result rank-metric codes.