[48,24,12]

codes/classical/bits/cyclic/golay.yml

@@ -67,9 +67,9 @@ relations:
     - code_id: nearly_perfect
       detail: 'The extended Golay code is nearly perfect.'
     - code_id: self_dual
-      detail: 'The extended Golay code is self-dual.'
+      detail: 'The extended Golay code is the unique code at its parameters and happens to be self-dual \cite[Remark 4.3.11]{preset:HPRainsSloane}.'
     - code_id: group
-      detail: 'The extended Golay code is a group-algebra code for various groups \cite{doi:10.1109/TIT.2008.928260,doi:10.1007/s10623-017-0440-7,doi:10.1016/0097-3165(90)90069-9}\cite[Ex. 16.5.1]{preset:HKSalgebra}.'
+      detail: 'The extended Golay code is a group-algebra code for various groups \cite{doi:10.1109/TIT.2008.928260,doi:10.1007/s10623-017-0440-7,doi:10.1016/0097-3165(90)90069-9}; see \cite[Ex. 16.5.1]{preset:HKSalgebra}.'
     - code_id: univ_opt_q-ary
       detail: 'The Golay code and several of its extended, shortened, and punctured versions are LP universally optimal codes \cite{arxiv:1212.1913}.'
 

codes/classical/bits/cyclic/self_dual_48_24_12.yml

@@ -0,0 +1,34 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: self_dual_48_24_12
+physical: bits
+logical: bits
+
+name: '\([48,24,12]\) self-dual code'
+
+description: |
+  An extended quadratic-residue code that is known to be the only self-dual doubly-even code at its parameters \cite{doi:10.1109/TIT.2002.806146}.
+
+  The code's automorphism group is \(PSL_2(47)\) \cite[Remark 4.3.11]{preset:HPRainsSloane}.
+
+relations:
+  parents:
+    - code_id: binary_quad_residue
+    - code_id: self_dual
+      detail: 'The \([48,24,12]\) self-dual code is the only self-dual doubly-even code at its parameters \cite{doi:10.1109/TIT.2002.806146}.'
+    - code_id: divisible
+      detail: 'The \([48,24,12]\) self-dual code is the only self-dual doubly-even code at its parameters \cite{doi:10.1109/TIT.2002.806146}.'
+  cousins:
+    - code_id: combinatorial_design
+      detail: 'Fixed-weight codewords of extremal self-dual doubly-even codes whose length divides 24 form a combinatorial 5-design \cite{doi:10.1016/S0021-9800(69)80115-8}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-03-15'

codes/classical/properties/block/symmetry/twisted/constacyclic.yml

@@ -13,6 +13,7 @@ alternative_names:
 
 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 \(\alpha c_n, c_1, \cdots, c_{n-1} \in C\).
+  A \(-1\)-constacyclic code is called \textit{negacyclic}.
 
 
 relations:

codes/classical/q-ary_digits/dual/self_dual.yml

@@ -32,6 +32,11 @@ protection: |
   except for \(n = 22\) modulo four for the second case, where the bound is increased by four \cite{doi:10.1109/18.651000}.
   A self-dual code saturating the above inequality is called \textit{extremal}.
 
+  Fixed-weight codewords of extremal self-dual doubly-even binary codes whose length divides 24 form a combinatorial 5-design \cite{doi:10.1016/S0021-9800(69)80115-8}.
+  The extended Golay code and the \([48,24,12]\) self-dual code are two such codes.
+  It is not yet known whether a \([72,36,16]\) self-dual code exists; see \cite{doi:10.1109/TIT.1973.1054975,doi:10.1090/conm/634/12692}\cite[Remark 4.3.11]{preset:HPRainsSloane}.
+
+  For ternary self-dual codes, see \cite[Remark 4.3.14]{preset:HPRainsSloane}\cite{doi:10.1016/j.ffa.2005.05.012}.
 
 notes:
   - See books \cite{doi:10.1007/3-540-30731-1,doi:10.1017/CBO9780511807077} for more on self-dual codes.
@@ -46,7 +51,7 @@ relations:
       In addition, quaternary linear codes are Hermitian self-orthogonal (self-dual) iff they are trace-Hermitian self-orthogonal (self-dual) additive \cite[Thm. 27.4.1]{preset:HKSquantum} (\cite[Thm. 9.10.3]{doi:10.1017/CBO9780511807077}).'
   cousins:
     - code_id: divisible
-      detail: 'Binary self-dual codes are singly-even and binary self-orthogonal codes that are not doubly-even are singly-even \cite[Def. 4.1.6]{preset:HKSselfdual}.
+      detail: 'Binary self-dual codes are singly-even, and binary self-orthogonal codes that are not doubly-even are singly-even \cite[Def. 4.1.6]{preset:HKSselfdual}.
       The minimum distance of doubly-even binary self-dual codes asymptotically satisfies \(d\leq0.1664n+o(n)\) \cite{doi:10.1109/18.605587}.'
 
 

codes/quantum/groups/group_quantum.yml

@@ -9,7 +9,16 @@ logical: groups
 
 name: 'Group-based quantum code'
 
-description: 'Encodes a \textit{logical} Hilbert space, finite- or infinite-dimensional, into a \textit{physical} Hilbert space of \(\ell^2\)-normalizable functions on a second-countable unimodular group. For \(K\)-dimensional logical subspace and for groups \(G^{n}\), can be denoted as \(((n,K))_G\). When the logical subspace is the Hilbert space of \(\ell^2\)-normalizable functions on \(G^{ k}\), can be denoted as \([[n,k]]_G\). Ideal codewords may not be normalizable, depending on whether \(G\) is continuous and/or noncompact, so approximate versions have to be constructed in practice.'
+description: 'Encodes a \textit{logical} Hilbert space, finite- or infinite-dimensional, into a \textit{physical} Hilbert space of \(\ell^2\)-normalizable functions on a second-countable unimodular group \(G\), i.e., a \(G\)\textit{-valued qudit}.
+For \(K\)-dimensional logical subspace and for block codes defined on groups \(G^{n}\), can be denoted as \(((n,K))_G\).
+When the logical subspace is the Hilbert space of \(\ell^2\)-normalizable functions on \(G^{ k}\), can be denoted as \([[n,k]]_G\).
+Ideal codewords may not be normalizable, depending on whether \(G\) is continuous and/or noncompact, so approximate versions have to be constructed in practice.'
+
+
+features:
+  general_gates:
+    - 'Various gates for a single \(G\)-valued qudit are described in \cite{arXiv:1911.00099,arxiv:2208.12309}.'
+
 
 relations:
   parents:

codes/quantum/properties/block/quantum_mds.yml

@@ -44,11 +44,11 @@ relations:
     - 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}.'
+      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: generalized_reed_solomon
       detail: 'Some MDS codes are constructed from cyclic and constacyclic codes \cite{arxiv:1502.05267} which are GRS codes \cite{doi:10.1007/s10623-022-01174-5,doi:10.1007/s10623-023-01294-6}.'
-    - 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}.'
+# - 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/qudits_galois/stabilizer/css/galois_css.yml

@@ -42,6 +42,8 @@ description: |
   |\gamma + C_Z^\perp \rangle = \frac{1}{\sqrt{|C_Z^\perp|}} \sum_{\eta \in C_Z^\perp} |\gamma + \eta\rangle.
   \end{align}
 
+  Galois-qudit CSS codes can also be understood in terms graphs via the \textit{reflexive stabilizer} framework, which also allows one to define a code for a given set of Pauli errors \cite{arxiv:2110.08414}.
+
 protection: 'Detects errors on \(d-1\) qubits, corrects errors on \(\left\lfloor (d-1)/2 \right\rfloor\) qubits.'
 
 relations: