weakly self-dual css code defn

codes/quantum/qubits/stabilizer/qubit_css.yml

@@ -36,7 +36,7 @@ description: |
   The resulting CSS code has \(k=k_X+k_Z-n\) logical qubits and distance \(d\geq\min\{d_X,d_Z\}\).
   The \(H_X\) (\(H_Z\)) block of \(H\) \eqref{eq:parity} is the parity-check matrix of the code \(C_Z\) (\(C_X\)).
   The requirement \(C_X^\perp \subseteq C_Z\) guarantees \eqref{eq:comm} and also implies  \(C_Z^\perp \subseteq C_X \).
-  Specializing to the case when \(C_Z=[n,k,d]\) is dual-containing yields an \([[n,2k-n,\geq d_Z]]\) qubit CSS code with \(C_X = C_Z^\perp\).
+  Specializing to the case when \(C_Z=[n,k,d]\) is dual-containing yields an \([[n,2k-n,\geq d_Z]]\) \textit{weakly self-dual qubit CSS code} with \(C_X = C_Z^\perp\).
   Basis states for the code are, for \(\gamma \in C_X\),
   \begin{align}
   |\gamma + C_Z^\perp \rangle = \frac{1}{\sqrt{|C_Z^\perp|}} \sum_{\eta \in C_Z^\perp} |\gamma + \eta\rangle.

codes/quantum/qudits/stabilizer/qudit_css.yml

@@ -33,6 +33,7 @@ description: |
   For composite \(q\), such codes need not encode an integer number of qudits, but the relation to homology allows for a general structure theorem \cite[Thm. 8.1.1]{arxiv:2405.03559}.
   For prime \(q=p\), properties reminiscent of qubit CSS codes are restored: encoding is based on two related \hyperref[code:q-ary_linear]{\(p\)-ary linear codes}, an \([n,k_X,d_X]_p \) code \(C_X\) and \([n,k_Z,d_Z]_p \) code \(C_Z\),
   satisfying \(C_X^\perp \subseteq C_Z\). The resulting CSS code has \(k=k_X+k_Z-n\) logical qubits and distance \(d\geq\min\{d_X,d_Z\}\).
+  Specializing to the case when \(C_Z=[n,k,d]_p\) is dual-containing yields an \([[n,2k-n,\geq d_Z]]_p\) \textit{weakly self-dual prime-qudit CSS code} with \(C_X = C_Z^\perp\).
   The \(H_X\) (\(H_Z\)) block of \(H\) \eqref{eq:parityq} is the parity-check matrix of the code \(C_X\) (\(C_Z\)). The requirement \(C_X^\perp \subseteq C_Z\) guarantees \eqref{eq:commQ}.
   Basis states for the code are, for \(\gamma \in C_X\),
   \begin{align}

codes/quantum/qudits_galois/stabilizer/css/galois_css.yml

@@ -37,7 +37,7 @@ description: |
   satisfying \(C_X^\perp \subseteq C_Z\).
   The resulting CSS code has \(k=k_X+k_Z-n\) logical Galois qudits and distance \(d\geq\min\{d_X,d_Z\}\).
   The \(H_X\) (\(H_Z\)) block of \(H\) \eqref{eq:parityg} is the parity-check matrix of the code \(C_X\) (\(C_Z\)). The requirement \(C_X^\perp \subseteq C_Z\) guarantees \eqref{eq:commG}.
-  Specializing to the case when \(C_Z=[n,k,d]_q\) is dual-containing yields a \([[n,2k-n,\geq d_Z]]_q\) Galois-qudit CSS code with \(C_X = C_Z^\perp\).
+  Specializing to the case when \(C_Z=[n,k,d]_q\) is dual-containing yields a \([[n,2k-n,\geq d_Z]]_q\) \textit{weakly self-dual Galois-qudit CSS code} with \(C_X = C_Z^\perp\).
   Basis states for the code are, for \(\gamma \in C_X\),
   \begin{align}
   |\gamma + C_Z^\perp \rangle = \frac{1}{\sqrt{|C_Z^\perp|}} \sum_{\eta \in C_Z^\perp} |\gamma + \eta\rangle.