self_complemetary

codes/classical/bits/binary_linear.yml

@@ -26,7 +26,7 @@ protection: |
   The decision problem corresponding to finding the minimum distance is also \(NP\)-complete \cite{doi:10.1109/18.641542}, and approximating the weight enumerator is \(\#P\)-complete \cite{arxiv:cs/0304044}.
 
 features:
-  rate: 'A family of linear codes \(C_i = [n_i,k_i,d_i]\) is \textit{asymptotically good} if the asymptotic rate \(\lim_{i\to\infty} k_i/n_i\) and asymptotic distance \(\lim_{i\to\infty} d_i/n_i\) are both positive. Nearly all good linear binary codes for the asymmetric channel are also good for the symmetric channel \cite{manual:{Varshamov, R. R. "Some features of linear codes that correct asymmetric errors." Soviet Physics Doklady. Vol. 9. 1965.}}.'
+  rate: 'A family of linear codes \(C_i = [n_i,k_i,d_i]\) is \textit{asymptotically good} if the asymptotic rate \(\lim_{i\to\infty} k_i/n_i\) and asymptotic distance \(\lim_{i\to\infty} d_i/n_i\) are both positive. Nearly all good linear binary codes for the asymmetric channel are also good for the symmetric channel \cite{manual:{Varshamov, R. R. "Some features of linear codes that correct asymmetric errors." Soviet Physics Doklady. Vol. 9. 1965.}}; this is not the case for non-binary codes \cite{arxiv:1310.7536}.'
 
   decoders:
     - 'Decoding an arbitary linear binary code is \(NP\)-complete \cite{doi:10.1109/TIT.1978.1055873}.'

codes/classical/bits/nonlinear/constantin_rao.yml

@@ -14,8 +14,8 @@ introduced: '\cite{doi:10.1016/S0019-9958(79)90329-2}'
 description: |-
   A nonlinear single-asymmetric-error code that generalize VT codes and that is constructed from an Abelian group.
 
-  A CR code for group \(G\) and fixed group element \(g\) consists of all binary strings \(c=c_1c_2\cdots c_n\) that satisfy \(\sum_i c_i g_i = g\) for some elements \(g_i\) \cite[Def. 1.3]{arxiv:1310.7536}.
-  Here, addition is the group operation, the multiplication \(1 g_i = g_i\), and \(0 g_i = 0_G\) where \(0_G\) is the identity element. 
+  A CR code for an Abelian group \(G\) of order \(n+1\) and fixed group element \(g\) consists of all binary strings \(c=c_1c_2\cdots c_n\) that satisfy \(\sum_{i=1}^n c_i g_i = g\) \cite[Def. 1.3]{arxiv:1310.7536}.
+  Here, addition is the group operation, the multiplication \(1 g_i = g_i\), and \(0 g_i = g_0\) is the identity element. 
 
   CR codes can be generalized to the \(q\)-ary case and also to codes correcting more than one asymmetric error \cite{manual:{Kløve, Torleiv. Error correcting codes for the asymmetric channel. Department of Pure Mathematics, University of Bergen, 1981.}}.
 
@@ -32,6 +32,10 @@ features:
 relations:
   parents:
     - code_id: bits_into_bits
+  cousins:
+    - code_id: q-ary_digits_into_q-ary_digits
+      detail: 'CR codes, and their special cases the VT codes, can be converted to ternary codes with nice structure via a \textit{binary-to-ternary} map \(00\to 0\), \(11\to 0\), \(01\to 1\), and \(10\to 2\) \cite{arxiv:1310.7536}.'
+
 
 
 # Begin Entry Meta Information

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

@@ -37,7 +37,7 @@ protection: |
   Other types of weight enumerators includes the Hamming weight enumerator, Lee weight enumerator, joint weight enumerator, split weight enumerator, and biweight enumerator \cite{preset:MacSlo}.
   \end{defterm}
 
-  Noise channels include the symmetric noise channel, asymmetric noise channels \cite{manual:{Varshamov, R. R. "Some features of linear codes that correct asymmetric errors." Soviet Physics Doklady. Vol. 9. 1965.},doi:10.1109/TIT.1973.1054954,doi:10.1016/S0019-9958(79)90329-2,manual:{Kløve, Torleiv. Error correcting codes for the asymmetric channel. Department of Pure Mathematics, University of Bergen, 1981.}}, and insertion/deletion noise.
+  Noise channels include the symmetric noise channel, asymmetric noise channels \cite{manual:{Varshamov, R. R. "Some features of linear codes that correct asymmetric errors." Soviet Physics Doklady. Vol. 9. 1965.},doi:10.1109/TIT.1973.1054954,doi:10.1016/S0019-9958(79)90329-2,manual:{Kløve, Torleiv. Error correcting codes for the asymmetric channel. Department of Pure Mathematics, University of Bergen, 1981.},doi:10.1142/6400}, and insertion/deletion noise.
 
 
 features:

codes/quantum/qubits/dynamic/floquet/floquet_xyz_ruby.yml

@@ -13,7 +13,9 @@ introduced: '\cite{arxiv:2407.08566}'
 description: |
   Floquet code whose qubits are placed on vertices of a ruby lattice.
   Its weight-two check operators are placed on various edges.
-  Its ISG can be that of the 6.6.6 color code concatenated with a three-qubit repetition code.
+  One third of the time during its measurement schedule, its ISG is that of the 6.6.6 color code concatenated with a three-qubit repetition code.
+  Together, all ISGs generate the gauge group of the 3F subsystem code.
+# from Ref. \cite{arxiv:0908.4246}, whose logical operators .
 
 
 features:
@@ -28,11 +30,13 @@ relations:
     - code_id: floquet
   cousins:
     - code_id: triangular_color
-      detail: 'The ISG of the XYZ ruby Floquet code can be that of the 6.6.6 color code concatenated with a three-qubit repetition code.'
+      detail: 'One third of the time during its measurement schedule, the ISG of the XYZ ruby Floquet code is that of the 6.6.6 color code concatenated with a three-qubit repetition code.'
     - code_id: quantum_repetition
-      detail: 'The ISG of the XYZ ruby Floquet code can be that of the 6.6.6 color code concatenated with a three-qubit repetition code.'
+      detail: 'One third of the time during its measurement schedule, the ISG of the XYZ ruby Floquet code is that of the 6.6.6 color code concatenated with a three-qubit repetition code.'
     - code_id: hexagonal
       detail: 'The XYZ ruby Floquet code is defined on the ruby lattice.'
+    - code_id: subsystem_three_fermion
+      detail: 'Together, all ISGs of the XYZ ruby Floquet code generate the gauge group of the 3F subsystem code.'
 
 
 # Begin Entry Meta Information

codes/quantum/qubits/nonstabilizer/union_stabilizer/cws/ssw.yml

@@ -12,7 +12,7 @@ short_name: 'SSW'
 introduced: '\cite{arxiv:quant-ph/0701065,doi:10.1090/S0002-9947-07-04242-0}'
 
 description: |
-  A family of \(((n=4k+2l+3,M_{k,l},2))\) CWS codes, where \(M_{k,l} \approx 2^{n-2}(1-\sqrt{2/(\pi(n-1))})\),  whose codewords are superpositions of particular bitstrings and their complements.
+  A family of \(((n=4k+2l+3,M_{k,l},2))\) self-complementary CWS codes, where \(M_{k,l} \approx 2^{n-2}(1-\sqrt{2/(\pi(n-1))})\).
   For \(n \geq 11\), these codes have a logical subspace whose dimension is larger than that of the largest stabilizer code for the same \(n\) and \(d\).
   A subset of these codes can be augmented to yield codes with one higher logical dimension \cite{arxiv:0709.1780}.
 
@@ -21,10 +21,9 @@ relations:
   parents:
     - code_id: cws
       detail: 'SSW codes can be formulated as CWS codes \cite{arxiv:0708.1021,manual:{Cross, Andrew William. Fault-tolerant quantum computer architectures using hierarchies of quantum error-correcting codes. Diss. Massachusetts Institute of Technology, 2008.}}.'
+    - code_id: self_complementary
     - code_id: small_distance_quantum
   cousins:
-    - code_id: iceberg
-      detail: 'SSW and \([[2m,2m-2,2]]\) codewords are superpositions of particular bitstrings and their complements.'
     - code_id: rains
       detail: 'The SSW code outperforms the Rains codes in terms of code parameters at odd \(n > 11\) \cite{arxiv:0708.1021,manual:{Cross, Andrew William. Fault-tolerant quantum computer architectures using hierarchies of quantum error-correcting codes. Diss. Massachusetts Institute of Technology, 2008.}}.'
 

codes/quantum/qubits/self_complementary.yml

@@ -0,0 +1,37 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: self_complementary
+physical: qubits
+logical: qubits
+
+
+name: 'Self-complementary quantum code'
+introduced: '\cite{arxiv:quant-ph/0701065,arxiv:0712.2586}'
+
+description: |
+  A qubit code which admits a basis of codewords of the form \(|c\rangle+|\overline{c}\rangle\), where \(c\) is a bitstring and \(\overline{c}\) is its negation a.k.a. complement. 
+  Their codewords generalize the two-qubit Bell states and three-qubit GHz states and are often called \textit{(qubit) cat states} or \textit{poor-man's GHz states}.
+  Such codes were originally pointed out to perform well against amplitude damping \cite{arxiv:0712.2586}.
+
+
+protection: |
+  Self-complementary codes automatically protect against a single \(Z\) error \cite{arxiv:quant-ph/0701065}, and can protect against a single amplitude damping error \cite{arxiv:0907.5149}.
+
+
+relations:
+  parents:
+    - code_id: qubits_into_qubits
+  cousins:
+    - code_id: linear_binary
+      detail: 'A linear binary code is called \textit{self-complementary} if, for each codeword \(c\), its negation \(\overline{c}\) is also a codeword.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-07-14'

codes/quantum/qubits/small_distance/iceberg.yml

@@ -15,12 +15,11 @@ alternative_names:
   - 'Iceberg code'
 
 description: |
-  CSS stabilizer code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits.
+  Self-complementary CSS code for \(m\geq 2\) with generators \(\{XX\cdots X, ZZ\cdots Z\} \) acting on all \(2m\) physical qubits.
   The code is constructed via the CSS construction from an SPC code and a repetition code \cite[Sec. III]{arxiv:1803.06987}.
   This is the highest-rate distance-two code when an even number of qubits is used \cite{arxiv:quant-ph/9608006}.
 
   Admits a basis such that each codeword is a superposition of a computational basis state labeled by an even-weight bitstring \(b\) and a state labeled by the negation of \(b\).
-  Such states generalize the two-qubit Bell states and three-qubit GHz states and are often called \textit{(qubit) cat states} or \textit{poor-man's GHz states}.
   Its all-zero logical state is a conventional GHz state.
 
   All of its automorphisms lie in the Clifford group \cite[Thm. 13]{arxiv:quant-ph/9704043}.
@@ -59,6 +58,7 @@ relations:
       detail: 'The \([[2m,2m-2,2]]\) error-detecting code forms one of the two families qubit quantum MDS codes \cite{arxiv:quant-ph/0312164}.'
     - code_id: ball_color
       detail: 'The \([[2m,2m-2,2]]\) error-detecting code is a ball color code \cite[Sec. III.A]{arxiv:2112.01446}.'
+    - code_id: self_complementary
   cousins:
     - code_id: parity_check
       detail: 'The \([[2m,2m-2,2]]\) error-detecting code is constructed via the CSS construction from an SPC code and its dual repetition code \cite[Sec. III]{arxiv:1803.06987}.'

codes/quantum/qudits_galois/stabilizer/qldpc/lp/matrix/expander_lifted_product.yml

@@ -21,9 +21,7 @@ description: |
 protection: 'Code performance strongly depends on \(G\). Certain non-Abelian groups yield asymptotically good QLDPC codes with parameters \([[n, k = \Theta(n), d = \Theta(n)]]\) \cite{arxiv:2111.03654}. Abelian groups like \(\mathbb{Z}_{\ell}\) for \(\ell=\Theta(n / \log n)\) yield constant-rate codes with parameters \([[n, k = \Theta(n), d = \Theta(n / \log n)]]\) \cite{arxiv:2012.04068}; this construction can be derandomized by being reformulated as a balanced product code \cite{arxiv:2012.09271}.'
 
 features:
-  rate: 'Expander lifted-product codes for non-Abelian groups include the first examples \cite{arxiv:2111.03654} of (asymptotically) \textit{good QLDPC codes}, i.e., codes with asymptotically constant rate and linear distance.
-  Expander LP codes for Abelian groups like \(\mathbb{Z}_{\ell}\) for \(\ell=\Theta(n / \log n)\) yield constant-rate codes with parameters \([[n, k = \Theta(n), d = \Theta(n / \log n)]]\) \cite{arxiv:2012.04068}; this construction can be derandomized by being reformulated as a balanced product code \cite{arxiv:2012.09271}.
-  Other explicit versions of codes with such parameters have been developed \cite{arxiv:2112.01647}.'
+  rate: 'Expander lifted-product codes for non-Abelian groups include the first examples \cite{arxiv:2111.03654} of (asymptotically) \textit{good QLDPC codes}, i.e., codes with asymptotically constant rate and linear distance. Expander LP codes for Abelian groups like \(\mathbb{Z}_{\ell}\) for \(\ell=\Theta(n / \log n)\) yield constant-rate codes with parameters \([[n, k = \Theta(n), d = \Theta(n / \log n)]]\) \cite{arxiv:2012.04068}; this construction can be derandomized by being reformulated as a balanced product code \cite{arxiv:2012.09271}.Other explicit versions of codes with such parameters have been developed \cite{arxiv:2112.01647}.'
 
   decoders:
     - 'Linear-time decoder \cite{arxiv:2206.07571}.'