# Explicit binary codes with block length n, distance n / log n, rate 1 - o(1)?

What are the best (i.e., highest-rate) explicit binary codes with block length $$n$$ and minimum distance $$d$$, in the regime $$d = n^{1 - o(1)}$$?

The "redundancy" of a code is the difference between the block length and message length. Nonexplicitly, it's easy to show that there exist binary codes with distance $$d$$, block length $$n$$, and redundancy $$O(d \log(n/d))$$. When $$d \leq n^{1 - \Omega(1)}$$, the nonexplicit bound is $$O(d \log n)$$, which can be matched explicitly using e.g. Reed-Solomon codes. When $$d = \alpha n$$ with $$\alpha$$ a sufficiently small positive constant, the nonexplicit bound is $$(1 - \Omega(1)) \cdot n$$, which can be matched explicitly using e.g. Justesen codes.

What about in the middle? E.g., what if $$d = n / \log n$$? The nonexplicit redundancy bound is $$o(n)$$. Is there an explicit code matching it up to a constant?