constacyclic + refs

codes/classical/properties/block/cyclic/constacyclic.yml

@@ -0,0 +1,25 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: constacyclic
+
+name: 'Constacyclic code'
+# introduced: '\cite{arXiv:math/0604603}'
+
+description: |
+  A classical code \(C\) of length \(n\) over an alphabet \(R\) is \(\alpha\)-constacyclic (or \(\alpha\)-twisted) if, for each string \(c_1 c_2 \cdots c_n\in C\), the string \(\alpha c_n c_1 \cdots c_{n-1} \in C\).
+
+
+relations:
+  parents:
+    - code_id: block
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2023-12-18'

codes/classical/properties/block/cyclic/cyclic.yml

@@ -15,6 +15,8 @@ relations:
   parents:
     - code_id: quasi_cyclic
       detail: 'Quasi-cyclic codes with \(\ell=1\) are cyclic.'
+    - code_id: constacyclic
+      detail: 'Constacyclic codes with \(\alpha=1\) are cyclic.'
     - code_id: group_orbit
       detail: 'All codewords of a cyclic code can be obtained from any codeword via cyclic shifts, meaning that the code consists of only one orbit.'
 

codes/classical/properties/block/cyclic/skew_cyclic.yml

@@ -10,9 +10,9 @@ introduced: '\cite{arXiv:math/0604603}'
 
 description: 'A classical code \(C\) of length \(n\) over an alphabet \(R\) is skew-cyclic if there exists an automorphism, \(\theta\), of \(R\), such that for each string \(c_1 c_2 \cdots c_n\in C\), the skew-cyclically shifted string \(\theta(c_n) \theta(c_1) \cdots \theta(c_{n-1})\in C\). We say that \(C\) is a \(\theta\)-cyclic code over \(R\).'
 
-features:
-  decoders:
-    - 'Only given for skew-BCH codes, adapted froom standard BCH codes.'
+# features:
+#   decoders:
+#     - 'Only given for skew-BCH codes, adapted froom standard BCH codes.'
 
 realizations:
   - 'Not directly implemented, but BCH codes form a subclass, and are used in DVD, solid state drive storage, etc.'
@@ -23,7 +23,7 @@ notes:
 relations:
   parents:
     - code_id: quasi_cyclic
-      detail: 'Under certain conditions, there is an equivalent quasi-cyclic code for every skew-cyclic code \cite{doi:10.1504/IJICOT.2011.044674}.'
+      detail: 'Under certain conditions, there is an equivalent quasi-cyclic  or cyclic code for every skew-cyclic code \cite{doi:10.1504/IJICOT.2011.044674}.'
 
 
 # Begin Entry Meta Information

codes/quantum/properties/block/quantum_mds.yml

@@ -41,6 +41,12 @@ relations:
     - code_id: mds
     - code_id: q-ary_cyclic
       detail: 'Quantum MDS codes can be constructed from \(q\)-ary cyclic codes using the Hermitian construction \cite{doi:10.1109/TIT.2011.2159039}.'
+    - code_id: stabilizer_over_gfqsq
+      detail: 'Many MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction \cite{arxiv:quant-ph/0312164,arxiv:0906.2509,arxiv:1507.08355,arxiv:1803.07927}, in particular from cyclic \cite{doi:10.1109/TIT.2011.2159039}, constacyclic \cite{doi:10.1109/TIT.2014.2308180,doi:10.1109/TIT.2015.2388576,arxiv:1803.07927} and negacyclic \cite{doi:10.1109/TIT.2012.2220519} codes.'
+    - code_id: constacyclic
+      detail: 'Many MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction from constacyclic codes \cite{doi:10.1109/TIT.2014.2308180,doi:10.1109/TIT.2015.2388576,arxiv:1803.07927}.'
+    - code_id: cyclic
+      detail: 'Many MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction \cite{arxiv:quant-ph/0312164,arxiv:0906.2509,arxiv:1507.08355,arxiv:1803.07927}, in particular from cyclic codes \cite{doi:10.1109/TIT.2011.2159039}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/small_distance/quantum_repetition.yml

@@ -51,7 +51,8 @@ realizations:
   - 'Semiconductor spin-qubit devices: 3-qubit devices at RIKEN \cite{arXiv:2201.08581} and Delft \cite{arXiv:2202.11530}.'
   - 'Nitrogen-vacancy centers in diamond: 3-qubit phase-flip code \cite{arXiv:1309.6424,doi:10.1038/s42005-022-00875-6} (see also Ref. \cite{arXiv:1309.5452}).'
   - 'Trapped-ion device: 3-qubit phase-flip algorithm implemented in 3 cycles on high fidelity gate operations \cite{doi:10.1126/science.1203329}.
-  Both phase- and bit-flip codes for 31 qubits and their stabilizer measurements have been realized by Quantinuum \cite{arxiv:2305.03828}.'
+  Both phase- and bit-flip codes for 31 qubits and their stabilizer measurements have been realized by Quantinuum \cite{arxiv:2305.03828}.
+  Multiple rounds of Steane error correction \cite{arxiv:2312.09745}.'
 
 #  - 'See Table S6 in Ref. \cite{arXiv:2102.06132} for a history of earlier implementations.'
 #  - 'Molecular spin qubits: 3-qubit molecule \cite{doi:10.1039/d0sc03107k}.'

codes/quantum/qubits/small_distance/small/steane.yml

@@ -42,7 +42,7 @@ description: |
 protection: 'The Steane code is a distance 3 code. It detects errors on 2 qubits, corrects errors on 1 qubit.'
 
 realizations:
-  - 'Trapped-ion qubits: seven-qubit device in Blatt group \cite{arXiv:1403.5426}, ten-qubit QCCD device by Quantinuum \cite{arXiv:2107.07505} realizing repeated rounds of error correction, real-time look-up-table decoding, and non-fault-tolerant magic-state distillations (see APS Physics Synopsis \cite{doi:10.1103/Physics.14.184}). Fault-tolerant universal two-qubit gate set by Monz group \cite{arxiv:2111.12654}. Logical CNOT gate between two logical qubits, including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 20-qubit device by Quantinuum \cite{arxiv:2208.01863}; logical fidelity interval of the combined preparation-CNOT-measurement procedure was higher than that of the unencoded physical qubits.'
+  - 'Trapped-ion qubits: seven-qubit device in Blatt group \cite{arXiv:1403.5426}, ten-qubit QCCD device by Quantinuum \cite{arXiv:2107.07505} realizing repeated rounds of error correction, real-time look-up-table decoding, and non-fault-tolerant magic-state distillations (see APS Physics Synopsis \cite{doi:10.1103/Physics.14.184}). Fault-tolerant universal two-qubit gate set by Monz group \cite{arxiv:2111.12654}. Logical CNOT gate between two logical qubits, including rounds of correction and fault-tolerant primitives such as flag qubits and pieceable fault tolerance, on a 20-qubit device by Quantinuum \cite{arxiv:2208.01863}; logical fidelity interval of the combined preparation-CNOT-measurement procedure was higher than that of the unencoded physical qubits. Multiple rounds of Steane error correction \cite{arxiv:2312.09745}.'
   - 'Rydberg atom arrays: Lukin group \cite{arXiv:2112.03923}; transversal CNOT gate performed on distance \(3\), \(5\), and \(7\) codes, logical ten-qubit GHZ state initialized with break-even fidelity, fault-tolerant logical two-qubit GHZ state initialized \cite{arxiv:2312.03982}.'
 
 features:

codes/quantum/qubits/stabilizer/qubit_css.yml

@@ -92,7 +92,7 @@ protection: |
 
   The distance of the CSS code is equal to the minimum of the combinatorial (\(d-1\))-systole of the cellulated \(d\)-dimensional manifold and its dual.
 
-  CSS codes have a \textit{CSS lower bound} against depolarizing noise, quantified by lower bounds on independently decoding the two classical codes \cite{doi:10.1109/ISIT.2013.6620358}.
+  CSS codes have a \textit{CSS lower bound} against depolarizing noise because CSS decoding does not take into account correlations between \(X\)- and \(Z\)-type noise \cite{doi:10.1109/ISIT.2013.6620358}.
 
 features:
   rate: 'For a depolarizing channel with probability \(p\), CSS codes allowing for arbitrarily accurate recovery exist with asymptotic rate \(1-2h(p)\), where \(h\) is the binary entropy function \cite{arxiv:quant-ph/0110143}.'

codes/quantum/qubits/stabilizer/stabilizer_over_gf4.yml

@@ -22,6 +22,7 @@ description: |
 
 protection: |
   A Hermitian self-orthogonal linear \([n,k,d]_{4}\) code yields an \([[n,n-2k]]\) qubit stabilizer code with distance no less than \(d\); this is the \textit{qubit Hermitian construction}.
+  A variant construction allows for the use of nearly self-orthogonal codes \cite{doi:10.1007/s10623-014-9934-8}.
 
 notes:
   - 'Tables of \([[n,0,d]]\) codes, corresponding to a self-dual \(GF(4)\) representation, at \href{http://www.ii.uib.no/~larsed/vncorbits/}{this website}.'
@@ -33,6 +34,8 @@ relations:
   cousins:
     - code_id: dual
       detail: 'Hermitian qubit codes are constructed from Hermitian self-orthogonal linear codes over \(GF(4)\) via the \hyperref[topic:gf4-representation]{\(GF(4)\) representation}.'
+    - code_id: constacyclic
+      detail: 'Duadic constacyclic codes yield many examples of Hermitian qubit codes \cite{arxiv:2312.06504}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/stabilizer/topological/color/color.yml

@@ -29,7 +29,6 @@ features:
 
   decoders:
     - 'Projection decoder \cite{doi:10.7907/059V-MG69}.'
-    - 'Matching decoder gives low logical failure rate \cite{arXiv:2108.11395}.'
     - 'Integer-program-based decoder \cite{arXiv:1402.3037}.'
     - 'Restriction decoder \cite{arxiv:1911.00355}.'
     - 'Cellular-automaton decoder for the \(XYZ\) color code \cite{arxiv:2203.16534}.'

codes/quantum/qubits/stabilizer/topological/color/triangular_color.yml

@@ -30,6 +30,8 @@ protection: |
 # In general, the definition \(\widehat{P} = P^{\otimes V}\) doesn’t guarantee transversality, because though \(\widehat{P}Z_f\widehat{P}Z_f = Z_f\), \(\widehat{P}X_f\widehat{P} = (-1)^{\frac{t}{2}}X_fZ_f\), where \(t\) is the number of vertices of \(f\). Therefore, \(t\) must be a multiple of 4. This is only true for the triangular color code defined on the 4-8-8 lattice.\cite{arXiv:1311.0879}'
 
 features:
+  decoders:
+    - 'Mobius matching decoder gives low logical failure rate \cite{arXiv:2108.11395} and has an open-source implementation \cite{arxiv:2312.08813}.'
 
   code_capacity_threshold:
     - '\(12.6\%\) threshold for triangular color code with the restriction decoder \cite{arXiv:1911.00355} and the projection decoder \cite{arXiv:1308.6207,arXiv:1802.08680}.'

codes/quantum/qudits_galois/stabilizer/galois_true_stabilizer.yml

@@ -34,8 +34,6 @@ relations:
   cousins:
     - code_id: q-ary_linear
       detail: 'A true Galois-qudit stabilizer code is the closest quantum analogue of a linear code over \(GF(q)\) because the \(q\)-ary vectors corresponding to the symplectic representation of the stabilizers form a linear subspace.'
-#    - code_id: constacyclic
-#      detail: 'The stabilizer-over-\(GF(q^2)\) can yield quantum MDS codes from Hermitian self-orthogonal constacyclic codes \cite{arxiv:1803.07927}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qudits_galois/stabilizer/stabilizer_over_gfqsq.yml

@@ -41,8 +41,6 @@ relations:
   cousins:
     - code_id: dual
       detail: 'Hermitian codes are constructed from Hermitian self-orthogonal linear codes over \(GF(q^2)\) via the \hyperref[topic:gfqsq-representation]{\(GF(q^2)\) representation}.'
-    - code_id: quantum_mds
-      detail: 'Many MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction \cite{arxiv:quant-ph/0312164,arxiv:0906.2509,arxiv:1507.08355,arxiv:1803.07927}, in particular from cyclic \cite{doi:10.1109/TIT.2011.2159039}, constacyclic \cite{doi:10.1109/TIT.2014.2308180,doi:10.1109/TIT.2015.2388576} and negacyclic \cite{doi:10.1109/TIT.2012.2220519} codes.'
     - code_id: matrix_product
       detail: 'Hermitian self-orthogonal matrix-product codes over \(GF(q^2)\) can be used to construct quantum codes via the Hermitian construction \cite{doi:10.1007/s11128-020-02921-0,arxiv:1604.05823}.'
     - code_id: galois_subsystem_stabilizer