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Oct 3, 2021 at 15:51 comment added Hhan I also used a similar approach, and I think that the same bound holds for most of the regime of $s$. To prove the statement for all $s$, it requires the approximation of the ratio of the number of Hamming sphere with radius $r$ and the number of points that has the exact Hamming weight $r$, up to the additive small error, which I failed to find. I think there should be relevant literature, though.
Oct 3, 2021 at 10:56 vote accept Marcel Dall'Agnol
Oct 2, 2021 at 13:19 history answered mathworker21 CC BY-SA 4.0