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 \(\Theta(N)\), and distances \[
d_X=\Theta(N),
\qquad
d_Z=\Theta(N^{1/m}). \] Moreover, the \(Z\)-stabilizer space has an explicit generating set of weight \(O(N^{1/m})\).
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 \(\Theta(n/(\log n)^4)\), with distances \[
d_X=\Omega\!\left(\frac{n}{(\log n)^4}\right),
\qquad
d_Z=\Omega\!\left(\frac{n^{1/m}}{(\log n)^{4/m}}\right), \] and the \(Z\)-stabilizer generating set remains sublinear.