Optimal rate–distance trade-off for binary codes

Determine the exact asymptotic trade-off between rate R and relative distance δ for binary codes over the field F2; that is, characterize the supremum of achievable rates R as a function of δ (equivalently, of δ as a function of R) for families of subsets of F2^n as n→∞.

Background

The paper studies low-rate binary codes and their distance, focusing on the Gilbert–Varshamov (GV) bound. While GV guarantees existence of codes with a certain rate–distance trade-off, the exact optimal trade-off for binary codes remains unknown. Establishing this fundamental curve would resolve a long-standing question in coding theory and contextualize progress on explicit and randomized constructions.

References

For binary codes (that is, codes where Σ = F_2), it is a major open question to pin down the best trade-off possible between rate and distance.

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

Thus the hierarchy is finitely convergent, while leaving open the distinct question of whether a bounded level already yields a sharp asymptotic exponent.

— The Honeycomb Framework for Code Bounds  (2608.20287 - Gay et al., 20 Aug 2026) in Section 6, subsection “The Three Directions of the Hierarchy”

More recently, constructs randomized and explicit families of asymptotically good CSS codes at any prescribed constant rate. Their encoders and inverse encoders use only CNOT gates, have $O(\log n)$ depth, and contain $O(n)$ gates to achieve linear distance. However, their exact rate-distance tradeoff remains undetermined, and they explicitly ask for a characterization of this tradeoff.

— Good Stabilizer Codes from Shallow Clifford Circuits with Random Matchings  (2608.18536 - Anand et al., 19 Aug 2026) in Section 1, Introduction