Cubical Sheaf Complexes with Constant Expansion with Applications to Asymptotically Good qLTCs

Abstract: For every fixed integers $r \ge 4$ and $2 \le k \le r-2$, we construct $r$-dimensional cubical sheaf complexes whose degree-$k$ CSS codes have positive constant rate, linear distance, and constant soundness, with bounded row and column weights. Taking $r=4$ and $k=2$ gives a family of asymptotically good binary qLTCs. At the core of our construction is a uniform product-expansion theorem for explicit Reed-Solomon codes on norm-one evaluation sets. The key point is that the expansion constant stays bounded away from zero as the local code lengths grow. We place these codes on arithmetic cubical complexes, obtaining constant local expansion for both the resulting sheaf and its dual. Together with the local-to-global framework of Dinur, Lin, and Vidick (FOCS 2024) and sheaf duality, this gives linear distance and constant soundness, while an asymmetric choice of local code dimensions gives positive rate. The resulting codes are explicit and polynomial-time computable.