group kingdom

codes/classical/groups/group_classical.yml

@@ -11,8 +11,6 @@ name: 'Group-alphabet code'
 description: |
   Encodes \(K\) states (codewords) in coordinates labeled by elements of a group \(G\). The number of codewords may be infinite for infinite groups, so various restricted versions have to be constructed in practice.
 
-protection: |
-  Bounds for permutation codes, i.e., codes on the symmetric group \(G=S_n\), have been developed \cite{doi:10.1016/0097-3165(79)90012-8,doi:10.1006/eujc.1998.0272}.
 
 relations:
   parents:

codes/classical/groups/group_linear.yml

@@ -10,17 +10,17 @@ name: 'Linear code over \(G\)'
 introduced: '\cite{doi:10.1109/49.29613,doi:10.1109/18.133243,doi:10.1109/18.104333}'
 
 description: |
-  Encodes \(K\) states (codewords) in \(n\) coordinates over a group \(G\) such that the codewords form a subgroup of \(G^n\). In other words, the set of codewords is closed under the group operation.
+  Block code that encodes \(K\) states (codewords) in \(n\) coordinates over a group \(G\) such that the codewords form a subgroup of \(G^n\). In other words, the set of codewords is closed under the group operation.
 
-  The \textit{automorphism group} of such codes is the group \(G^n\) formed by the action of \(G\) on each coordinate as well as the coordinate permutation group \(S_n\).
+  For finite groups that are not finite fields, the \textit{automorphism group} of such codes is the group \(G^n\) formed by the action of \(G\) on each coordinate as well as the coordinate permutation group \(S_n\).
 
 relations:
   parents:
     - code_id: group_classical
     - code_id: group_orbit
-      detail: 'The set of codewords of a linear code over \(G\) can be thought of as an orbit of a particular codeword under the group formed by the code.
-      However, group orbit codes do not have to be linear \cite[Remark 8.4.3]{preset:EricZin}.'
+      detail: 'The set of codewords of a linear code over \(G\) can be thought of as an orbit of a particular codeword under the group formed by the code. However, group orbit codes do not have to be linear \cite[Remark 8.4.3]{preset:EricZin}.'
     - code_id: block
+      detail: 'Linear codes over \(G\) are linear block codes with \(\Sigma=G\).'
   cousins:
     - code_id: group_gkp
 

codes/classical/groups/mixed/binary-ternary.yml

@@ -4,7 +4,6 @@
 #######################################################
 
 code_id: binary-ternary
-logical: groups
 physical: groups
 
 name: 'Binary-ternary mixed code'
@@ -16,7 +15,7 @@ protection: |
   See Ref. \cite{doi:10.1109/18.651001,arxiv:1606.06930} for bounds on binary-ternary mixed codes.
 
 notes:
-  - 'Binary-ternary mixed codes have been used in football pools, in which \(n_1\) of the matches result in either a win, a loss, or a draw, but \(n_2\) of the matches are assumed to have only two of the three possible outcomes \cite{doi:10.1016/0097-3165(91)90024-B}.'
+  - 'Binary-ternary mixed codes have been used in football pools, in which \(n_1\) of the matches result in either a win, a loss, or a draw, but \(n_2\) of the matches are assumed to have only a win or a loss outcome \cite{doi:10.1016/0097-3165(91)90024-B}.'
 
 
 relations:

codes/classical/groups/mixed/mixed.yml

@@ -9,7 +9,10 @@ physical: groups
 
 name: 'Mixed code'
 
-description: 'Encodes \(K\) states (codewords) in a string of coordinates which takes values in more than one group.'
+alternative_names:
+  - 'Mixed-alphabet code'
+
+description: 'Encodes \(K\) states (codewords) in a string of two or more coordinates, each of which takes values in one of two or more possible groups.'
 
 protection: |
   The Hamming, Singleton, and Plotkin bounds are straightforwardly extended to mixed alphabets \cite[Thm. 5.1]{manual:{Cameron, Peter J. "Some bridges between codes and designs." Unpublished manuscript, Queen Mary and Westfield College, London (1998).}}.
@@ -19,7 +22,7 @@ relations:
     - code_id: group_classical
   cousins:
     - code_id: orthogonal_array
-      detail: 'Orthogonal arrays generalized to mixed alphabets are caled mixed-level orthogonal arrays \cite{manual:{Addelman, S., & Kempthorne, O. (1961b). Orthogonal Main-Effect Plans. Technical Report ARL 79, Aeronautical Research Lab., Wright-Patterson Air Force Base, Ohio, Nov. 1961.},doi:10.1016/B978-0-7204-2262-7.50034-X}, (see \cite[Ch. 9]{doi:10.1007/978-1-4612-1478-6}). See Ref. \cite{doi:10.1016/S0378-3758(96)00025-0} for bounds on mixed orthogonal arrays.'
+      detail: 'Orthogonal arrays generalized to mixed alphabets are called mixed-level orthogonal arrays \cite{manual:{Addelman, S., & Kempthorne, O. (1961b). Orthogonal Main-Effect Plans. Technical Report ARL 79, Aeronautical Research Lab., Wright-Patterson Air Force Base, Ohio, Nov. 1961.},doi:10.1016/B978-0-7204-2262-7.50034-X}, (see \cite[Ch. 9]{doi:10.1007/978-1-4612-1478-6}). See Ref. \cite{doi:10.1016/S0378-3758(96)00025-0} for bounds on mixed orthogonal arrays.'
     - code_id: combinatorial_design
       detail: 'Combinatorial designs have been generalized to mixed alphabets \cite{doi:10.1023/A:1008340128973}.'
 

codes/classical/groups/binary_permutation.yml → codes/classical/groups/permutation/binary_permutation.yml

@@ -5,24 +5,32 @@
 
 code_id: binary_permutation
 physical: groups
-logical: q-ary_digits
 
-name: 'Binary permutation-based code'
-introduced: '\cite{doi:10.1016/S0019-9958(79)90076-7}'
+name: 'Code in permutations'
+introduced: '\cite{doi:10.1109/TIT.1969.1054291,doi:10.1016/S0019-9958(79)90076-7}'
+
+alternative_names:
+  - 'Permutation-based code'
 
 description: |
   Encodes codewords into permutations of \(n\) objects.
 # If this is a perm rep, this it should be faithful so there should be group codes.
 
+protection: |
+  Protects against errors in the Kendall tau distance on the space of permutations.
+  The Kendall distance between permutations \(\sigma\) and \(\pi\) is defined as the minimum number of adjacent transpositions required to change \(\sigma\) into \(\pi\).
+  Various bounds have been developed \cite{doi:10.1016/0097-3165(79)90012-8,doi:10.1006/eujc.1998.0272}.
+
 notes:
   - 'Review of parallels between linear binary codes and permutation groups \cite{doi:10.1016/j.ejc.2009.03.044}.'
 
 relations:
   parents:
     - code_id: group_classical
+      detail: 'Codes in permutations are group-alphabet codes for the symmetric group \(G=S_n\).'
   cousins:
     - code_id: convolutional
-      detail: 'Permutation convolutional codes have been constructed \cite{doi:10.1109/TCOMM.2005.858683}.'
+      detail: 'Convolutional codes in permutations have been constructed \cite{doi:10.1109/TCOMM.2005.858683}.'
 
 
 # Begin Entry Meta Information

codes/classical/groups/permutation/rank_modulation.yml

@@ -0,0 +1,40 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: rank_modulation
+physical: groups
+
+name: 'Rank-modulation code'
+introduced: '\cite{doi:10.1109/ISIT.2008.4595285,arxiv:1110.2557}'
+
+description: |
+  A family of codes in permutations derived from \(q\)-ary linear codes, such as Lee-metric codes, RS codes \cite{arxiv:1110.2557}, quadratic residue codes, and most binary codes.
+
+features:
+  rate: 'Rank modulation codes with code distance of \hyperref[topic:asymptotics]{order} \(d=\Theta(n^{1+\epsilon})\) for \(\epsilon\in[0,1]\) achieve a rate of \(1-\epsilon\) \cite{doi:10.1109/ISIT.2010.5513604}.'
+
+realizations:
+  - 'Electronic devices where charges can either increase in an individual cell or decrease in a block of adjacent cells, e.g., flash memories \cite{doi:10.1109/TIT.2009.2018336}.'
+
+relations:
+  parents:
+    - code_id: binary_permutation
+  cousins:
+    - code_id: gray
+      detail: 'The rank-modulation Gray code is an extension of the original binary Gray code to a code on the permutation group \cite{doi:10.1109/TIT.2009.2018336}.'
+    - code_id: q-ary_linear
+      detail: 'Almost all linear \(q\)-ary codes can be converted to rank-modulation codes \cite{arxiv:1110.2557}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2022-11-07'
+    - user_id: VictorVAlbert
+      date: '2022-04-12'
+    - user_id: JiaxinHuang
+      date: '2022-04-08'

codes/classical/properties/block/block.yml

@@ -71,9 +71,6 @@ description: |
   In this way, the field elements form the Klein four group \(\mathbb{Z}_2\times\mathbb{Z}_2\) under addition.
   One can check that the trace operation, \(\text{tr}(\gamma) = \gamma + \gamma^2\), outputs either 0 or 1 for any element \(\gamma\in GF(4)\).
 
-  Two \(q\)-ary codes are \textit{equivalent} if the codewords of one code can be mapped into those of the other under a combination of a coordinate permutation and a permutation of the elements of each coordinate. 
-  The full group of such composite permutations is \(S_q \wr S_n\) \cite[Def. 1.8.8]{preset:HKSbasics}\cite{preset:HKSclass}.
-
 # The field trace in this case is similar to taking the real part of a complex number, \(\text{tr}(\gamma) = \gamma + \gamma^2\) for any element \(\gamma\).
 
 # The group of isometries of \(q\)-ary Hamming space is a combination of the monomial group, the Galois group of \(GF(q)\), and the group formed by the action of the space on itself (under addition) \cite{preset:HKSclass}.

codes/classical/q-ary_digits/q-ary_digits_into_q-ary_digits.yml

@@ -11,9 +11,14 @@ name: '\(q\)-ary code'
 
 description: |
   Encodes \(K\) states (codewords) in \(n\) \(q\)-ary coordinates over the field \(GF(q)\), i.e., \(q\)-ary strings.
-  Their error-correcting performance is quantified by some distance \(d\), which in turn is defined using a metric.
+  Error-correcting performance is quantified by some distance \(d\), which in turn is defined using a metric.
   The default distance is the Hamming distance \(d\), the weight (i.e., number of nonzero coordinates) of the lowest-weight nonzero codeword; such codes are usually denoted as \((n,K,d)_q\).
   The corresponding Hamming metric between two \(q\)-ary strings is the number of coordinates in which they differ.
+  Unless stated otherwise, the distance for this class is the Hamming distance.
+
+  Two \(q\)-ary codes are \textit{equivalent} if the codewords of one code can be mapped into those of the other under a combination of a coordinate permutation and a permutation of the elements of each coordinate. 
+  The full group of such composite permutations is \(S_q \wr S_n\) \cite[Def. 1.8.8]{preset:HKSbasics}\cite{preset:HKSclass}.
+
 
 protection: |
   A code detects errors on up to \(d-1\) coordinates, corrects erasure errors on up to \(d-1\) coordinates, and corrects general errors on up to \(\left\lfloor (d-1)/2 \right\rfloor\) coordinates.

codes/classical/rings/rings_into_rings.yml

@@ -13,6 +13,7 @@ description: 'Encodes \(K\) states (codewords) in \(n\) coordinates over a finit
 relations:
   parents:
     - code_id: block
+      detail: 'Ring codes are block codes with \(\Sigma=R\).'
     - code_id: ecc_finite
     - code_id: group_classical
       detail: 'A ring \(R\) is an Abelian group under addition.'

codes/quantum/qubits/small_distance/small/stab_10_1_2.yml

@@ -11,11 +11,11 @@ name: '\([[10,1,2]]\) CSS code'
 introduced: '\cite{arxiv:2112.01446}'
 
 description: |
-  Smallest stabilizer code to implement a transversal \(T\) gate.
+  Smallest stabilizer code to implement a logical \(T\) gate via application of physical \(T\), \(T^{\dagger}\), and \(CCZ\) gates.
 
 features:
   transversal_gates:
-    - 'Logical \(T\) gate via application of \(T\), \(T^{\dagger}\), and \(CCZ\) \cite{arxiv:2112.01446}.'
+    - 'Logical \(T\) gate via application of physical \(T\), \(T^{\dagger}\), and \(CCZ\) gates \cite{arxiv:2112.01446}.'
 
   general_gates:
     - 'Magic-state distillation protocol \cite{arxiv:2112.01446}.'

codes/quantum/qubits/stabilizer/topological/color/3d_color/stab_15_1_3.yml

@@ -29,8 +29,7 @@ features:
   magic_scaling_exponent: 'Magic-state yield parameter \( \gamma= \log_d (n/k)\approx 2.47\) \cite[Box 2]{arxiv:1612.07330}\cite{arxiv:1703.07847}.'
 
   transversal_gates:
-    - 'This is the smallest qubit stabilizer code with a (strongly) transversal gate outside of the \hyperref[topic:clifford]{Clifford group} \cite{arxiv:2210.14066}.'
-    - 'A transversal logical \(T^\dagger\) is implemented by applying a \(T\) gate on every qubit \cite{arxiv:quant-ph/9610011,arxiv:1403.2734,arxiv:1612.07330}.'
+    - 'A transversal logical \(T\) is implemented by applying a \(T^\dagger\) gate on every qubit \cite{arxiv:quant-ph/9610011,arxiv:1403.2734,arxiv:1612.07330}. This is the smallest qubit stabilizer code with a (strongly) transversal gate outside of the \hyperref[topic:clifford]{Clifford group} \cite{arxiv:2210.14066}.'
     - 'A subsystem version yields a transversal \(CCZ\) gate \cite{arxiv:1304.3709}.'
 
   general_gates: