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).

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., 2024) in Section 1 (Introduction)

On one hand, the most likely avenue for answering the above question about QLDPC codes is also via probabilistic construction since we do not know of explicit codes matching the GV bound even without the LDPC property (or even classically!).

— Random Quantum LDPC Codes Approaching the Gilbert-Varshamov Bound  (2610.02648 - Mittal et al., 2 Oct 2026) in Section 1, Introduction