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codes/classical/bits/ta-shma.yml

@@ -11,7 +11,7 @@ name: 'Ta-Shma zigzag code'
 introduced: '\cite{doi:10.1145/3055399.3055408}'
 
 description: |
-  Member of a family of \(\epsilon\)-balanced codes that nearly achieves the \hyperref[topic:gv-bound]{asymptotic GV bound}. The codes have relative distance \(\frac{1}{2}-\frac{\epsilon}{2}\) and rate of order \(\Omega (\epsilon^{2+\beta})\) for \(\beta\to 0\) as \(n\to\infty\) \cite{arxiv:2011.05500}.
+  Member of a family of \(\epsilon\)-balanced codes that nearly achieves the \hyperref[topic:gv-bound]{asymptotic GV bound}. The codes have relative distance \(\frac{1}{2}-\frac{\epsilon}{2}\) and rate of \hyperref[topic:asymptotics]{order} \(\Omega (\epsilon^{2+\beta})\) for \(\beta\to 0\) as \(n\to\infty\) \cite{arxiv:2011.05500}.
 
 features:
   decoders:

codes/quantum/properties/block/block_quantum.yml

@@ -26,6 +26,7 @@ protection: |
   \subsection{Bounds on code parameters}
   Bounds on finite dimensional block code performance include the quantum Singleton bound, quantum Hamming bound, \hyperref[topic:quantum-gv-bound]{quantum GV bound}, various quantum linear programming (LP) bounds \cite{arxiv:quant-ph/9611001,arxiv:quant-ph/9709049} (see the book \cite{preset:GottesmanBook}), and other bounds \cite{doi:10.1109/TIT.2005.862086,arxiv:1007.3655}.
   A code whose parameters attain the quantum Hamming bound (quantum Singleton bound) is called a perfect quantum code (a quantum MDS code).
+  We are often interested in how parameters of particular infinite block quanutm code families scale with increasing block length \(n\), necessitating the use of \hyperref[topic:asymptotics]{asymptotic notation}.
 
   \begin{defterm}{Quantum GV bound}
   \label{topic:quantum-gv-bound}

codes/quantum/qubits/stabilizer/topological/surface/hyperbolic/freedman_meyer_luo.yml

@@ -11,13 +11,18 @@ name: 'Freedman-Meyer-Luo code'
 introduced: '\cite{doi:10.1201/9781420035377-13}'
 
 description: |
-  Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom \cite{doi:10.2140/gtm.1999.2.113}. Codes derived from such manifolds can achieve distances scaling better than \(\sqrt{n}\), something that is impossible using closed 2D surfaces or 2D surfaces with boundaries \cite{doi:10.1063/1.4726034}. Improved codes are obtained by studying a weak family of Riemann metrics on closed 4-dimensional manifolds \(S^2\otimes S^2\) with the \(Z_2\)-homology.
+  Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom \cite{doi:10.2140/gtm.1999.2.113}. Codes derived from such manifolds can achieve distances scaling better than \(\sqrt{n}\), something that is impossible using closed 2D surfaces or 2D surfaces with boundaries \cite{doi:10.1063/1.4726034}. Improved codes are obtained by studying a weak family of Riemann metrics on closed 4-dimensional manifolds \(S^2\otimes S^2\) with the \(Z_2\)-homology. 
+  
+  The Freedman-Meter-Luo code has been generalized to a family with rate of \hyperref[topic:asymptotics]{order} \(O(1/\sqrt{\log n})\) and minimum distance of \hyperref[topic:asymptotics]{order} \(\Omega(\sqrt{\log n})\) which supports fault-tolerant non-Clifford gates \cite{arxiv:2310.16982}.
 
 protection: 'Four-dimensional manifolds with weak systolic freedom yield \([[n,2,\Omega(\sqrt{n \sqrt{\log n}})]]\) surface codes.'
 
 features:
   rate: 'Codes held a 20-year record the best lower bound on asymptotic scaling of the minimum code distance, \(d=\Omega(\sqrt{n \sqrt{\log n}})\), broken by Ramanujan tensor-product codes.'
 
+  fault_tolerance:
+    - 'The Freedman-Meter-Luo code has been generalized to a family with rate of \hyperref[topic:asymptotics]{order} \(O(1/\sqrt{\log n})\) and minimum distance of \hyperref[topic:asymptotics]{order} \(\Omega(\sqrt{\log n})\) which supports fault-tolerant non-Clifford gates \cite{arxiv:2310.16982}.'
+
 notes:
   - 'See thesis by Fetaya for pedagogical exposition \cite{arxiv:1108.2886}.'