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Explicit constructions approaching the GV bound at non-vanishing rates

Develop explicit constructions of binary codes over F2 that approach the Gilbert–Varshamov bound while having rates bounded away from zero (i.e., a fixed positive rate independent of block length).

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Background

The GV bound applies across rates, but most progress on explicit binary constructions near GV has been in the very low-rate regime. Achieving codes that approach GV at rates bounded away from zero remains unresolved and would significantly broaden the applicability of explicit near-capacity constructions.

References

However, there are still open questions. For example, we do not know how to attain δ = \frac{1 - \epsilon}{2} and R = \Omega(\epsilon2) (without any o(1) term) explicitly, and we do not have explicit constructions approaching the GV bound with rates bounded away from zero.

— When Do Low-Rate Concatenated Codes Approach The Gilbert-Varshamov Bound? (2405.08584 - Doron et al., 14 May 2024) in Section 1 (Introduction)