  A major goal for fault-tolerant quantum computation (FTQC) is to reduce the
overhead needed for error correction. One approach is to use block codes that
encode multiple qubits, which can achieve significantly higher rates for the
same code distance than single-qubit code blocks or topological codes. We
present a scheme for universal quantum computation using multi-qubit
Calderbank-Shor-Steane (CSS) block codes, where codes admitting different
transversal gates are used to achieve universality, and logical teleportation
is used to move qubits between code blocks. All circuits for both computation
and error correction are transversal. We also argue that single shot
fault-tolerant error correction can be done in Steane syndrome extraction.
Then, we present estimates of information lifetime for a few possible codes,
which suggests that highly nontrivial quantum computations can be achieved at
reasonable error rates, using codes that require significantly less than 100
physical qubits per logical qubit.
