Asymptotically good divisible codes with asymptotically good duals

Determine whether, for every fixed integer \(\ell \ge 3\), there exists a family of asymptotically good \(2^\ell\)-divisible binary codes whose dual codes are also asymptotically good.

Background

The paper’s constructions for ℓ>1\ell>1 use punctured Reed–Muller codes as auxiliary CSS codes. These auxiliary codes provide transversal implementations of logical single-qubit ZZ-rotations, but their logical dimension and distance scale only sublinearly with blocklength. The authors explain that asymptotically good CSS families supporting the corresponding logical transversal would improve the parameters of their constructions for arbitrary target-support sizes.

The stated classical coding problem asks for asymptotically good 2ℓ2^\ell-divisible codes whose duals are simultaneously asymptotically good. An affirmative answer would directly resolve the associated quantum coding problem and, according to the paper, would yield asymptotically good CSS families for arbitrary support sizes.

References

Solving the following classical coding problem, initially posed in for $\ell=3$ ($8$-divisible codes, see), would directly resolve this quantum open problem. For any fixed $\ell \ge 3$, does there exist a family of asymptotically good $2\ell$-divisible codes whose dual codes are also asymptotically good?

— Realizing Logical Diagonal Gates via Transversal Physical $Z$-Rotations in CSS Codes  (2608.19094 - Reddy et al., 19 Aug 2026) in Open Problem 1, Section “CSS code families for arbitrary target vector support size”