Quantum Codes with Transversal $CCZ$ Gates and Sublinear $Z$-Stabilizers

We construct asymmetric quantum CSS codes with transversal $CCZ$ gates from algebraic expander codes \cite{KT26}. For every fixed $m\ge 3$, our growing-alphabet codes have length $N$, dimension $\mathrm{\Theta }(N)$, and distances $$d_X=\mathrm{\Theta }(N),d_Z=\mathrm{\Theta }(N^{1\mathrm{/}m})\mathrm{.}$$ Moreover, the $Z$-stabilizer space has an explicit generating set of weight $O(N^{1\mathrm{/}m})$. We build on the algebraic puncturing framework of Golowich and Guruswami \cite{GG24}, which turns classical codes with the required Schur-product and distance conditions into CSS codes with transversal $CCZ$. However, applying the framework directly to the algebraic expander codes runs into their small dual distance, and therefore produces only sublinear dimension. Our main technical step is a refined puncturing theorem in which the global dual-distance assumption is replaced by a condition only on the selected puncturing set. We also reduce the alphabet to a fixed prime field using a projective-multiplicity version of multiplication-friendly codes. The resulting fixed-prime-field CSS code triples, of length $n$, still have transversal $CCZ$ gates. Their dimension is $\mathrm{\Theta }(n\mathrm{/}(\log n{)}^4)$, with distances $$d_X=\mathrm{\Omega }\text{\hspace{0.17em}⁣}\left(\frac{n}{(\log n{)}^4}\right),d_Z=\mathrm{\Omega }\text{\hspace{0.17em}⁣}\left(\frac{n^{1\mathrm{/}m}}{(\log n{)}^{4\mathrm{/}m}}\right),$$ and the $Z$-stabilizer generating set remains sublinear.