A fermionic code related to the exceptional group E-8

In this paper we study the Hamming-like fermionic code encoding three-qubits into 16 Majorana modes, recently introduced by Hastings. We show that although this fermionic code cannot be obtained from a single qubit stabilizer code via the usual procedure, it can be obtained from two qubit stabilizer codes via a glucing procedure combining both single and double occupancy embeddings of qubits into fermionic Fock space. This technique identifies the code subspace as a Cartan subspace of the largest exceptional group E-8. In arriving at these results we develop a general formalism for implementing this glueing procedure via the use of intertwiners between different subsectors of the Fock space realizing embedded qubits.