small_distance

errorcorrectionzoo · Nov 23, 2023 · 30e8b50 · 30e8b50
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diff --git a/codelists/descendants/classical/list_small.yml b/codelists/descendants/classical/list_small.yml
@@ -0,0 +1,31 @@
+
+list_id: 'small'
+
+title: |
+  Small-distance classical codes
+
+intro: |
+  Here is a list of block codes of length \(n\) that detect or correct an error on at most two coordinates.
+
+
+display:
+  style: table
+  options:
+    cssclass: xtra-xtra-stretch
+    columns:
+      - property: name
+        title: Code
+        link_to_code: true
+        cssclass: 'top left'
+
+      - property: description
+        title: Description
+        cssclass: 'textit'
+        first_paragraph_only: true
+
+sort:
+  by: name
+
+codes:
+  select:
+    - descendant_of: small_distance
diff --git a/...lists/descendants/quantum/list_single.yml → ...escendants/quantum/list_small_quantum.yml b/...lists/descendants/quantum/list_single.yml → ...escendants/quantum/list_small_quantum.yml
@@ -1,8 +1,8 @@
 
-list_id: 'single'
+list_id: 'small_quantum'
 
 title: |
-  Small-distance codes
+  Small-distance quantum codes
 
 intro: |
   Here is a list of codes defined on \(n\) identical subsystems (e.g., qubits, modular qudits, or Galois qudits) that detect or correct an error on at most one subsystem.
@@ -28,4 +28,4 @@ sort:
 
 codes:
   select:
-    - descendant_of: small_distance
+    - descendant_of: small_distance_quantum
diff --git a/codes/classical/bits/cyclic/gold.yml b/codes/classical/bits/cyclic/gold.yml
@@ -10,7 +10,7 @@ logical: bits
 name: 'Gold code'
 introduced: '\cite{doi:10.1109/TIT.1967.1054048}'
 
-description: 'Member of the family of \([2^r-1, 2r ]\) cyclic binary linear code characterized by the generator polynomial of degree \(r\) of two maximum-period sequences of period \(2^r-1\) with absolute cross-correlation \( \leq 2^{(r+2)/2}\). Gold codewords are generated using \(m\)-sequences \(x\) and \(y\), which are codewords of simplex codes with check polynomials of degree \(r\) \cite{doi:10.1109/TIT.1967.1054048}.'
+description: 'Member of the family of \([2^r-1, 2r ]\) cyclic binary linear codes characterized by the generator polynomial of degree \(r\) of two maximum-period sequences of period \(2^r-1\) with absolute cross-correlation \( \leq 2^{(r+2)/2}\). Gold codewords are generated using \(m\)-sequences \(x\) and \(y\), which are codewords of simplex codes with check polynomials of degree \(r\) \cite{doi:10.1109/TIT.1967.1054048}.'
 
 features:
   encoders:

diff --git a/codes/classical/bits/cyclic/melas.yml b/codes/classical/bits/cyclic/melas.yml
@@ -20,6 +20,7 @@ relations:
   parents:
     - code_id: binary_cyclic
     - code_id: reversible
+    - code_id: small_distance
 # MacWilliams Sloane
   cousins:
     - code_id: quaternary_over_z4

diff --git a/codes/classical/bits/easy/hamming/extended_hamming.yml b/codes/classical/bits/easy/hamming/extended_hamming.yml
@@ -16,6 +16,7 @@ description: |
 relations:
   parents:
     - code_id: binary_linear
+    - code_id: small_distance
   cousins:
     - code_id: univ_opt_q-ary
       detail: 'Several extended Hamming codes are LP universally optimal codes \cite{arxiv:1212.1913}.'

diff --git a/codes/classical/bits/nonlinear/preparata.yml b/codes/classical/bits/nonlinear/preparata.yml
@@ -9,7 +9,7 @@ physical: bits
 logical: bits
 
 name: 'Preparata code'
-introduced: '\cite{doi:10.1007/0-387-34799-2_39}'
+introduced: '\cite{doi:10.1016/S0019-9958(68)90874-7}'
 
 description: |
   A nonlinear binary \((2^{m+1}-1, 2^{m+1}-2m-2, 5)\) code where \(m\) is odd.
@@ -36,6 +36,7 @@ relations:
   parents:
     - code_id: nearly_perfect
       detail: 'Preparata codes are uniformly packed and nearly perfect \cite{doi:10.1016/0012-365X(72)90025-8}. For any word \(u\) and Preparata codebook \(C\) with \(d(u, C) > 2\), we have that \(u\) has a distance 2 or 3 to exactly floor((2^{m+1}-1)/3) codewords.'
+    - code_id: small_distance
   cousins:
     - code_id: quaternary_over_z4
       detail: 'Preparata codes can be seen, via the Gray map, as linear codes over \(\mathbb{Z}_4\) \cite{doi:10.1109/18.312154,arxiv:math/0207208}.'

diff --git a/codes/classical/bits/nonlinear/sphere_packing/best.yml b/codes/classical/bits/nonlinear/sphere_packing/best.yml
@@ -19,6 +19,7 @@ description: |
 relations:
   parents:
     - code_id: bits_into_bits
+    - code_id: small_distance
   cousins:
     - code_id: construction_a
       detail: 'Using Construction \(A\), the Best code yields \(P_{10c}\), a non-lattice sphere packing in 10 dimensions that is the densest known \cite{doi:10.4153/CJM-1971-081-3}\cite[pg. 140]{doi:10.1007/978-1-4757-6568-7}.'

diff --git a/codes/classical/bits/nonlinear/sphere_packing/sloane_whitehead.yml b/codes/classical/bits/nonlinear/sphere_packing/sloane_whitehead.yml
@@ -20,7 +20,7 @@ relations:
   parents:
     - code_id: bits_into_bits
     - code_id: uplusv
-
+    - code_id: small_distance
 
 # Begin Entry Meta Information
 _meta:

diff --git a/codes/classical/bits/nonlinear/vasilyev.yml b/codes/classical/bits/nonlinear/vasilyev.yml
@@ -21,7 +21,7 @@ relations:
     - code_id: uplusv
     - code_id: perfect_binary
       detail: 'Vasilyev codes are perfect nonlinear binary codes and are inequivalent to any linear code.'
-
+    - code_id: small_distance
 
 # Begin Entry Meta Information
 _meta:

diff --git a/codes/classical/properties/block/small_distance.yml b/codes/classical/properties/block/small_distance.yml
@@ -0,0 +1,27 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: small_distance
+# includes Galois, modular, oscillators, and groups
+
+name: 'Small-distance block code'
+
+description: |
+  A block code of length \(n\) that either detects or corrects errors on at most two coordinates, i.e., has distance \(d \leq 5\).
+
+
+relations:
+  parents:
+    - code_id: block
+  cousins:
+    - code_id: small_distance_quantum
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2023-11-22'
diff --git a/codes/classical/q-ary_digits/easy/hexacode.yml b/codes/classical/q-ary_digits/easy/hexacode.yml
@@ -51,6 +51,7 @@ relations:
       detail: 'The hexacode is an MDS code \cite[Exer. 578]{doi:10.1017/CBO9780511807077}.'
     - code_id: extended_reed_solomon
       detail: 'The hexacode is an extended RS code \cite[pg. 82]{doi:10.1007/978-1-4757-6568-7}.'
+    - code_id: small_distance
   cousins:
     - code_id: q-ary_hamming
       detail: 'The hexacode is an extended ternary Hamming code \cite[Exer. 578]{doi:10.1017/CBO9780511807077}.'

diff --git a/codes/classical/q-ary_digits/easy/q-ary_hamming.yml b/codes/classical/q-ary_digits/easy/q-ary_hamming.yml
@@ -20,6 +20,7 @@ relations:
     - code_id: perfect
     - code_id: univ_opt_q-ary
       detail: 'Hamming codes and their punctured and shortened versions are LP universally optimal codes \cite{arxiv:1212.1913}.'
+    - code_id: small_distance
   cousins:
     - code_id: incidence_matrix
       detail: 'Columns of a Hamming parity-check matrix correspond to one-dimensional subspaces of \(GF(q)^n\).'

diff --git a/codes/classical/q-ary_digits/easy/q-ary_parity_check.yml b/codes/classical/q-ary_digits/easy/q-ary_parity_check.yml
@@ -18,6 +18,7 @@ relations:
     - code_id: q-ary_cyclic
       detail: 'Since permutations coordinate sums, the cyclic permutation of an SPC codeword is another codeword. The generator polynomial of the code is \(x-1\).'
     - code_id: mds
+    - code_id: small_distance
   cousins:
     - code_id: q-ary_ldgm
       detail: 'Concatenated \(q\)-ary parity-check codes are LDGM \cite{doi:10.1109/20.917609}.'

diff --git a/codes/classical/q-ary_digits/easy/ternary_golay.yml b/codes/classical/q-ary_digits/easy/ternary_golay.yml
@@ -55,6 +55,7 @@ relations:
       detail: 'The ternary Golay code is perfect.'
     - code_id: delsarte_optimal_q-ary
       detail: 'The ternary Golay code and one of its shortened versions are \(q\)-ary sharp configurations \cite[Table 12.1]{preset:HKSbounds}.'
+    - code_id: small_distance
   cousins:
     - code_id: golay
     - code_id: self_dual

diff --git a/codes/classical/rings/octacode.yml b/codes/classical/rings/octacode.yml
@@ -11,7 +11,7 @@ name: 'Octacode'
 introduced: '\cite{doi:10.1016/0097-3165(93)90070-O,doi:10.1007/978-1-4757-6568-7,arXiv:math/0208001}'
 
 description: |
-  The unique self-dual linear code of length 8 over \(\mathbb{Z}_4\) with generator matrix
+  The unique self-dual linear code of length 8 and Lee distance 6 over \(\mathbb{Z}_4\) with generator matrix
   \begin{align}
     \begin{pmatrix}
     3 & 3 & 2 & 3 & 1 & 0 & 0 & 0\\

diff --git a/codes/classical/rings/pentacode.yml b/codes/classical/rings/pentacode.yml
@@ -11,12 +11,13 @@ name: 'Pentacode'
 introduced: '\cite{doi:10.1007/bf01388558}'
 
 description: |
-  Nonlinear code over \(\mathbb{Z}_4\) whose codewords are all cyclic permutations and negations of the strings \(01112\), \(03110\), \(21310\), and \(21132\).
+  Nonlinear \((5,40,4)_{\mathbb{Z}_4}\) code over \(\mathbb{Z}_4\) whose codewords are all cyclic permutations and negations of the strings \(01112\), \(03110\), \(21310\), and \(21132\).
 
 
 relations:
   parents:
     - code_id: rings_into_rings
+    - code_id: small_distance
   cousins:
     - code_id: best
       detail: 'Codewords of the Best code can be obtained by applying the Gray map to the pentacode \cite[Sec. 2]{doi:10.1007/BF01388558}.'

diff --git a/codes/quantum/groups/group_4_2_2.yml b/codes/quantum/groups/group_4_2_2.yml
@@ -22,7 +22,7 @@ relations:
       detail: 'The four group-qudit code is the smallest quantum double code.'
     - code_id: covariant
       detail: 'The four group-qudit code is \(G\)-covariant.'
-    - code_id: small_distance
+    - code_id: small_distance_quantum
   cousins:
     - code_id: iceberg
       detail: 'The four group-qudit code can be extended to the \([[2m,2m-2,2]]_{G}\) group-qudit code \cite[Sec. VIII]{arxiv:1902.07714}. The latter reduces to the \([[2m,2m-2,2]]\) error-detecting code for \(G=\mathbb{Z}_2\).'

diff --git a/codes/quantum/groups/rotors/stabilizer/css/current_mirror.yml b/codes/quantum/groups/rotors/stabilizer/css/current_mirror.yml
@@ -33,7 +33,7 @@ features:
 relations:
   parents:
     - code_id: homological_rotor
-    - code_id: small_distance
+    - code_id: small_distance_quantum
   cousins:
     - code_id: gkp
       detail: 'Current-mirror code phase gates utilize ancillary osillators in square-lattice GKP states \cite{arXiv:cond-mat/0609441,arxiv:1302.4122}.'

diff --git a/codes/quantum/groups/rotors/stabilizer/css/rotor_3_1_2.yml b/codes/quantum/groups/rotors/stabilizer/css/rotor_3_1_2.yml
@@ -26,7 +26,7 @@ relations:
       detail: 'Taking \(H_X=\begin{pmatrix}-3 & 1 & 2\end{pmatrix}\) and \(H_Z=\begin{pmatrix}4&6&3\end{pmatrix}\) yields the three-rotor code.'
     - code_id: covariant
       detail: 'The three-rotor code is \(U(1)\)-covariant.'
-    - code_id: small_distance
+    - code_id: small_distance_quantum
   cousins:
     - code_id: stab_3_1_2
 

diff --git a/codes/quantum/groups/rotors/stabilizer/css/zero_pi.yml b/codes/quantum/groups/rotors/stabilizer/css/zero_pi.yml
@@ -47,7 +47,7 @@ notes:
 relations:
   parents:
     - code_id: homological_rotor
-    - code_id: small_distance
+    - code_id: small_distance_quantum
   cousins:
     - code_id: gkp
       detail: 'Zero-pi code phase gates utilize ancillary osillators in square-lattice GKP states \cite{arXiv:cond-mat/0609441,arxiv:1302.4122}.'

diff --git a/codes/quantum/groups/rotors/stabilizer/rotor_5_1_3.yml b/codes/quantum/groups/rotors/stabilizer/rotor_5_1_3.yml
@@ -22,7 +22,7 @@ relations:
     - code_id: covariant
       detail: 'The five-rotor code is \(U(1)\)-covariant.'
     - code_id: quantum_cyclic
-    - code_id: small_distance
+    - code_id: small_distance_quantum
   cousins:
     - code_id: qudit_5_1_3
       detail: 'The five-rotor code is a rotor analogue of the five-qudit code.'

diff --git a/codes/quantum/oscillators/stabilizer/hyperplane/braunstein.yml b/codes/quantum/oscillators/stabilizer/hyperplane/braunstein.yml
@@ -21,7 +21,7 @@ relations:
   parents:
     - code_id: analog_stabilizer
     - code_id: quantum_cyclic
-    - code_id: small_distance
+    - code_id: small_distance_quantum
   cousins:
     - code_id: qudit_5_1_3
       detail: 'The Braunstein five-mode code is a bosonic analogue of the five-qudit code.'

diff --git a/codes/quantum/oscillators/stabilizer/hyperplane/lloyd_slotine.yml b/codes/quantum/oscillators/stabilizer/hyperplane/lloyd_slotine.yml
@@ -35,7 +35,7 @@ realizations:
 relations:
   parents:
     - code_id: analog_stabilizer
-    - code_id: small_distance
+    - code_id: small_distance_quantum
   cousins:
     - code_id: shor_nine
       detail: 'The Lloyd-Slotine nine-mode code is a bosonic analogue of Shor''s code.'

diff --git a/...antum/properties/block/small_distance.yml → ...operties/block/quantum_small_distance.yml b/...antum/properties/block/small_distance.yml → ...operties/block/quantum_small_distance.yml
@@ -3,13 +3,13 @@
 ##       https://github.com/errorcorrectionzoo       ##
 #######################################################
 
-code_id: small_distance
+code_id: small_distance_quantum
 # includes Galois, modular, oscillators, and groups
 
 name: 'Small-distance block quantum code'
 
 description: |
-  A block code on \(n\) subsystems that either detects or corrects errors on only a single subsystem. These two cases correspond to distance \(d=2\) or \(d=3\) block quantum codes, respectively.
+  A block quantum code on \(n\) subsystems that either detects or corrects errors on only a single subsystem. These two cases correspond to distance \(d=2\) or \(d=3\) block quantum codes, respectively.
 
   All single error-correcting qubit stabilizer codes have been classified \cite{arxiv:0901.1968}. See Ref. \cite{arxiv:quant-ph/9704043} for single error-detecting qubit stabilizer codes.
 

diff --git a/codes/quantum/qubits/fermions/mbq.yml b/codes/quantum/qubits/fermions/mbq.yml
@@ -29,7 +29,7 @@ features:
 relations:
   parents:
     - code_id: majorana_stab
-    - code_id: small_distance
+    - code_id: small_distance_quantum
     - code_id: topological_abelian
       detail: 'When treated as ground states of the code Hamiltonian, surface codewords realize, codewords of a single Kitaev chain realize \(\mathbb{Z}_2\) fermionic topological order.
       The MZMs used to define the tetron code act as Ising anyons, which are non-Abelian.'

diff --git a/...ntum/qubits/nonstabilizer/permutation_invariant/binary_dihedral_permutation_invariant.yml b/...ntum/qubits/nonstabilizer/permutation_invariant/binary_dihedral_permutation_invariant.yml
@@ -30,7 +30,7 @@ relations:
     - code_id: j_gross
       detail: 'Binary dihedral permutation-invariant codes can be interpreted as Clifford single-spin codes.'
   cousins:
-    - code_id: small_distance
+    - code_id: small_distance_quantum
       detail: 'The first and second families of binary dihedral permutation-invariant codes have distance three.'
     - code_id: xp_stabilizer
       detail: 'Binary dihedral permutation invariant codewords form error spaces of XP stabilizer codes.'

diff --git a/.../quantum/qubits/nonstabilizer/permutation_invariant/icosahedral_permutation_invariant.yml b/.../quantum/qubits/nonstabilizer/permutation_invariant/icosahedral_permutation_invariant.yml
@@ -31,7 +31,7 @@ relations:
     - code_id: qubits_into_qubits
     - code_id: j_gross
       detail: 'The \(((7,2,3))\) permutation-invariant code can be interpreted as a spin-\(7/2\) Clifford code \cite{arXiv:2005.10910}.'
-    - code_id: small_distance
+    - code_id: small_distance_quantum
 
 
 # Begin Entry Meta Information

diff --git a/codes/quantum/qubits/rm/diagonal_clifford.yml b/codes/quantum/qubits/rm/diagonal_clifford.yml
@@ -21,7 +21,7 @@ features:
 relations:
   parents:
     - code_id: quantum_reed_muller
-    - code_id: small_distance
+    - code_id: small_distance_quantum
 
 # Begin Entry Meta Information
 _meta:

diff --git a/codes/quantum/qubits/small_distance/h_code_jones.yml b/codes/quantum/qubits/small_distance/h_code_jones.yml
@@ -28,7 +28,7 @@ features:
 relations:
   parents:
     - code_id: qubit_css
-    - code_id: small_distance
+    - code_id: small_distance_quantum
 
 
 # Begin Entry Meta Information

diff --git a/codes/quantum/qubits/small_distance/iceberg.yml b/codes/quantum/qubits/small_distance/iceberg.yml
@@ -44,7 +44,7 @@ relations:
     - code_id: qubit_css
     - code_id: quantum_mds
       detail: 'The \([[2m,2m-2,2]]\) error-detecting code is one of the two qubit quantum MDS codes \cite{arxiv:quant-ph/0312164}.'
-    - code_id: small_distance
+    - code_id: small_distance_quantum
 
 
 # Begin Entry Meta Information

diff --git a/codes/quantum/qubits/small_distance/quantum_hamming.yml b/codes/quantum/qubits/small_distance/quantum_hamming.yml
@@ -19,7 +19,7 @@ protection: 'Protects against any single qubit error.'
 relations:
   parents:
     - code_id: qubit_stabilizer
-    - code_id: small_distance
+    - code_id: small_distance_quantum
   cousins:
     - code_id: quantum_perfect
       detail: 'Quantum Hamming codes saturate the asymptotic quantum Hamming bound.'

diff --git a/codes/quantum/qubits/small_distance/small/qubit_5_6_2.yml b/codes/quantum/qubits/small_distance/small/qubit_5_6_2.yml
@@ -18,7 +18,7 @@ relations:
     - code_id: quantum_cyclic
     - code_id: non_stabilizer
       detail: 'The six-qubit CWS code is a non-stabilizer qubit code \cite{arxiv:quant-ph/0210097}.'
-    - code_id: small_distance
+    - code_id: small_distance_quantum
 
 
 # Begin Entry Meta Information

diff --git a/codes/quantum/qubits/small_distance/small/stab_8_3_2.yml b/codes/quantum/qubits/small_distance/small/stab_8_3_2.yml
@@ -33,7 +33,7 @@ relations:
   parents:
     - code_id: 3d_color
       detail: 'The \([[8,3,2]]\) code is the smallest non-trivial 3D color code.'
-    - code_id: small_distance
+    - code_id: small_distance_quantum
   cousins:
     - code_id: stab_15_1_3
       detail: 'The \([[8,3,2]]\) code can be obtained from a subset of physical qubits of the \([[15,1,3]]\) code \cite{arxiv:2112.01446}.'

diff --git a/codes/quantum/qubits/small_distance/tfim.yml b/codes/quantum/qubits/small_distance/tfim.yml
@@ -23,7 +23,7 @@ relations:
   parents:
     - code_id: qubit_stabilizer
     - code_id: translationally_invariant_stabilizer
-    - code_id: small_distance
+    - code_id: small_distance_quantum
   cousins:
     - code_id: majorana_stab
       detail: 'The TFIM code stabilizers can be expressed in terms of Majorana operators.'

diff --git a/codes/quantum/qubits/stabilizer/topological/surface/non-css/rhombic_dodecahedron_surface.yml b/codes/quantum/qubits/stabilizer/topological/surface/non-css/rhombic_dodecahedron_surface.yml
@@ -16,7 +16,7 @@ description: |
 relations:
   parents:
     - code_id: clifford-deformed_surface
-    - code_id: small_distance
+    - code_id: small_distance_quantum
   cousins:
     - code_id: polytope
       detail: 'The rhombic dodecahedron surface code arranges qubits and stabilizer generators on polytopes.'

diff --git a/codes/quantum/qubits/stabilizer/topological/surface/two_dim/stellated_dodecahedron_css.yml b/codes/quantum/qubits/stabilizer/topological/surface/two_dim/stellated_dodecahedron_css.yml
@@ -20,7 +20,7 @@ description: |
 relations:
   parents:
     - code_id: two_dimensional_hyperbolic_surface
-    - code_id: small_distance
+    - code_id: small_distance_quantum
   cousins:
     - code_id: polytope
       detail: 'Bring''s code and related codes listed in \cite[Table 1]{arXiv:1712.07666} arrange qubits and stabilizer generators on polytopes.'

diff --git a/codes/quantum/qubits/stabilizer/topological/surface/two_dim/surface-17.yml b/codes/quantum/qubits/stabilizer/topological/surface/two_dim/surface-17.yml
@@ -33,7 +33,7 @@ realizations:
 relations:
   parents:
     - code_id: rotated_surface
-    - code_id: small_distance
+    - code_id: small_distance_quantum
   cousins:
     - code_id: shor_nine
       detail: 'Both Shor''s code and surface-17 are \([[9,1,3]]\) codes, but they are distinct (e.g., they have different weight enumerators).'