  We discuss two methods to encode one qubit into six physical qubits. Each of
our two examples corrects an arbitrary single-qubit error. Our first example is
a degenerate six-qubit quantum error-correcting code. We explicitly provide the
stabilizer generators, encoding circuit, codewords, logical Pauli operators,
and logical CNOT operator for this code. We also show how to convert this code
into a non-trivial subsystem code that saturates the subsystem Singleton bound.
We then prove that a six-qubit code without entanglement assistance cannot
simultaneously possess a Calderbank-Shor-Steane (CSS) stabilizer and correct an
arbitrary single-qubit error. A corollary of this result is that the Steane
seven-qubit code is the smallest single-error correcting CSS code. Our second
example is the construction of a non-degenerate six-qubit CSS
entanglement-assisted code. This code uses one bit of entanglement (an ebit)
shared between the sender and the receiver and corrects an arbitrary
single-qubit error. The code we obtain is globally equivalent to the Steane
seven-qubit code and thus corrects an arbitrary error on the receiver's half of
the ebit as well. We prove that this code is the smallest code with a CSS
structure that uses only one ebit and corrects an arbitrary single-qubit error
on the sender's side. We discuss the advantages and disadvantages for each of
the two codes.
