Explicit low-rate construction achieving δ = (1 − ε)/2 and R = Ω(ε^2) without o(1) losses

Construct explicit families of binary linear codes over F2 with relative distance δ = (1 − ε)/2 and rate R ≥ c·ε^2 for some absolute constant c > 0 and all sufficiently small ε > 0, eliminating the current o(1) loss in the best-known explicit constructions.

Background

In the low-rate regime, the GV bound guarantees existence of binary linear codes with relative distance (1−ε)/2 and rate Ω(ε^2). Recent explicit constructions achieve δ = (1−ε)/2 with rate Ω(ε^{2+o(1)}), but the o(1) term persists. Removing this o(1) term to meet the GV bound exactly in the explicit setting would tighten the best-known constructions to match the existential benchmark.