Binarized-and-concatenated (B&C) phantom code[1]

Description

Member of a family of \(k=2\) CSS phantom codes obtained from a \(q=4\) Galois-qudit CSS code by binarizing each \(\mathbb{F}_4\) qudit into two qubits and then concatenating each qubit pair with the \([[4,2,2]]\) code [1].

The construction starts from a \([[n,1,d]]_4\) CSS code satisfying a condition that realizes the Frobenius transform \(\gamma\mapsto\gamma^2\) by a coordinate permutation; together with field multiplication \(\gamma\mapsto\alpha\gamma\), this supplies all \(\mathrm{GL}(2,\mathbb{F}_2)\) transformations on the two binary components of the logical \(\mathbb{F}_4\) qudit after concatenation. Binarization uses the self-dual normal basis \(\{\omega,\omega^2\}\) of \(\mathbb{F}_4\). The subsequent \([[4,2,2]]\) layer maps a single \(\mathbb{F}_4\)-qudit Pauli to four-qubit Paulis, e.g., \(X^\omega\mapsto XXII\), \(X^{\omega^2}\mapsto XIXI\), \(X^1\mapsto IXXI\), and similarly \(Z^{\omega^2}\mapsto IIZZ\), \(Z^\omega\mapsto IZIZ\), \(Z^1\mapsto IZZI\) [1].