Good binary quantum codes with transversal CCZ gate

We give an asymptotically good family of quantum CSS codes on qubits with a transversal CCZ gate, meaning that the parallel logical CCZ on all logical qubits is performed by parallel physical CCZs on (a subset of) physical qubits. Our construction is based on the observation that any classical code satisfying a multiplication property can be used to construct a quantum CSS code with transversal (qudit) CCZ. To obtain a constant-rate and linear-distance family, we instantiate this construction with a classical good family of algebraic-geometry codes on a non-binary, but constant-sized, alphabet. We then use a technique from the arithmetic secret sharing literature to reduce the alphabet to binary. As a corollary, our code family provides a constant-overhead magic state distillation scheme.