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.