constantin_rao

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.'
+  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.}}.'
 
   decoders:
     - 'Decoding an arbitary linear binary code is \(NP\)-complete \cite{doi:10.1109/TIT.1978.1055873}.'

codes/classical/bits/easy/hamming/hamming.yml

@@ -41,6 +41,8 @@ relations:
     - code_id: univ_opt_q-ary
       detail: 'Binary Hamming codes and several of their extended, punctured, and shortened versions are LP universally optimal codes \cite{arxiv:1212.1913}.'
   cousins:
+    - code_id: constantin_rao
+      detail: 'The nonlinear CR codes for \(G = \mathbb{Z}_2^r\) reduce to Hamming codes at lengths \(n = 2^r - 1\) \cite{manual:{Kløve, Torleiv. Error correcting codes for the asymmetric channel. Department of Pure Mathematics, University of Bergen, 1981.}}; see Ref. \cite{arxiv:1310.7536}.'
     - code_id: extended_hamming
     - code_id: reed_muller
       detail: 'Binary Hamming codes are equivalent to RM\(^*(r-2,r)\).'

codes/classical/bits/nonlinear/constantin_rao.yml

@@ -0,0 +1,42 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: constantin_rao
+physical: bits
+logical: bits
+
+name: 'Constantin-Rao (CR) code'
+short_name: 'CR'
+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 \(c=c_1c_2\cdotsc_n\) that satisfy \(\sum_i^n c_i g_i = g\) 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. 
+
+  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.}}.
+
+protection: |
+  Protect against single errors induced by the asymmetric noise channel.
+  Codes for some groups, and in particular, the VT codes, also protect against single deletions and insertions \cite{doi:10.1109/ITW.2009.5351208}.
+
+features:
+  rate: |
+    CR codes for particular groups have higher rates than single-error-correcting codes under the binary asymmetric channel for all lengths except \(n = 2^r - 1\), in which case CR codes reduce to Hamming codes \cite{manual:{Kløve, Torleiv. Error correcting codes for the asymmetric channel. Department of Pure Mathematics, University of Bergen, 1981.}}; see Ref. \cite{arxiv:1310.7536}.
+    Size analysis is presented in Ref. \cite{doi:10.1016/S0019-9958(80)90082-0}.'
+
+
+relations:
+  parents:
+    - code_id: bits_into_bits
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-07-13'

codes/classical/bits/nonlinear/vt_single_deletion.yml

@@ -7,17 +7,20 @@ code_id: vt_single_deletion
 physical: bits
 logical: bits
 
-name: 'Binary Varshamov-Tenengolts (VT) code'
-short_name: 'Binary VT'
+name: 'Varshamov-Tenengolts (VT) code'
+short_name: 'VT'
 introduced: '\cite{manual:{R. R. Varshamov and G. M. Tenengolts, \emph{Codes which correct single asymmetric errors} (translated to English), Autom. Remote Control, 26(2), 286-290 (1965)},manual:{V. I. Levenshtein, \emph{Binary codes capable of correcting deletions, insertions and reversals} (translated to English), Soviet Physics Dokl., 10(8), 707-710 (1966).}}'
 
 description: |-
-  Nearly optimal binary deletion-correcting code. Given integers \(n\geq 1\) and \(a\in\{0,1,\dots,n\}\), the associated binary Varshamov-Tenengolts code \(C_{n,a}\) corresponds to the set
-
+  Nearly optimal binary deletion-correcting code and code for the asymmetric channel. 
+  
+  Given integers \(n\geq 1\) and \(a\in\{0,1,\dots,n\}\), the associated Varshamov-Tenengolts code \(C_{n,a}\) corresponds to the set
   \begin{align}
   C_{n,a}=\left\{x\in\{0,1\}^n: \sum_{i=1}^n i~x_i = a\mod (n+1) \right\}.
   \end{align}
 
+  VT codes can be generalized to the \(q\)-ary case \cite{doi:10.1109/TIT.1984.1056962}.
+
 protection: 'Corrects a single asymmetric error (a \(0\) mapped to a \(1\)), a single deletion, or a single insertion of an arbitrary bit in an arbitrary position for any choice of \(a\).'
 
 features:
@@ -28,11 +31,12 @@ features:
 
 relations:
   parents:
-    - code_id: bits_into_bits
+    - code_id: constantin_rao
+      detail: 'CR codes for \(G=\mathbb{Z}_{n+1}\) reduce to VT codes.'
     - code_id: insertion_deletion
   cousins:
     - code_id: binary_linear
-      detail: 'By adapting a construction of Tenengolts \cite{manual:{G. M. Tenengolts, \emph{Class of codes correcting bit loss and errors in the preceding bit} (translated to English), Automation and Remote Control, 37(5), 797–802 (1976).}}, binary VT codes can be modified to yield slightly longer linear codes \cite{arxiv:math/0207197}.'
+      detail: 'By adapting a construction of Tenengolts \cite{manual:{G. M. Tenengolts, \emph{Class of codes correcting bit loss and errors in the preceding bit} (translated to English), Automation and Remote Control, 37(5), 797–802 (1976).}}, VT codes can be modified to yield slightly longer linear codes \cite{arxiv:math/0207197}.'
 
 
 # Begin Entry Meta Information

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

@@ -22,6 +22,7 @@ protection: |
     A code \(C\) corrects \(e\) deletions if all codewords are separated by at least \(e+1\) in the deletion distance \cite{arxiv:math/0207197}.
     Similar distances, collectively called \textit{editing distances}, can be defined for insertions and related operations \cite[Sec. 22.7]{preset:HKSmetrics}.
 
+
 # features:
 #   rate: 'For binary codes, the optimal achievable rate to protect against a \(\delta\) fraction of insertions and deletions is \(1- O(\delta \log\frac{1}{\delta})\).'
 # In the large alphabet setting, there exists a code which corrects a \(\delta\) fraction of insertions and deletions and has a rate of \((1-\Omega)^5\).'
@@ -33,6 +34,9 @@ notes:
 relations:
   parents:
     - code_id: q-ary_digits_into_q-ary_digits
+  cousins:
+    - code_id: permutation_invariant
+      detail: 'PI codes of distance \(d\) can protect against \(d-1\) (quantum) deletion errors.'
 
 
 # Begin Entry Meta Information

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

@@ -37,6 +37,8 @@ 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,manual:{Kløve, Torleiv. Error correcting codes for the asymmetric channel. Department of Pure Mathematics, University of Bergen, 1981.}}, and insertion/deletion noise.
+
 
 features:
   rate: The rate of a \(q\)-ary code is usually defined as \(R=\frac{1}{n}\log_q K\) dits per symbol.

codes/quantum/oscillators/fock_state/bosonic_q-ary_expansion.yml

@@ -20,6 +20,9 @@ description: |
       N=\sum_{\mu=0}^{\infty}b_{\mu}p^{\mu}~,
   \end{align}
   with each \(p\)-ary string \(b_{\mu}\in\mathbb{Z}_p\) corresponding to the basis-state label of qudit \(\mu\).
+  
+  In the binary case, the first qubit's \(Z\)-operator is the parity operator \(Z_0=(-1)^{\hat{n}}\), while the second qubit's \(Z\)-operator is the two-photon parity \(Z_1=(-1)^{\frac{1}{2}\hat{n}(\hat{n}-1)}\) \cite{arxiv:1106.3800,arxiv:2012.06994}. 
+  These satisfy \(Z_{1}aZ_{1}=aZ_{0}\).
 
   Pauli operators for the constituent qudits can be expressed in terms of bosonic raising and lowering operators.
   The modular-qudit Pauli-\(Z\) operator for qudit \(\mu\) is the Fock-space rotation

codes/quantum/properties/block/symmetric/permutation_invariant.yml

@@ -16,7 +16,7 @@ description: |
 #Such a code is said to be stabilized by the symmetric group \(S_n\) on \(n\) elements, in a generalization of stabilizer codes to binary codes utilizing (nonabelian) group actions.'
 
 protection: |
-  Permutation invariant codes of distance \(d\) can protect against \(d-1\) deletion errors \cite{arxiv:2001.08405,arxiv:2004.00814,arxiv:2102.02494,arxiv:2102.03015}, i.e., erasures of subsystems at unknown locations.
+  PI codes of distance \(d\) can protect against \(d-1\) deletion errors \cite{arxiv:2001.08405,arxiv:2004.00814,arxiv:2102.02494,arxiv:2102.03015}, i.e., erasures of subsystems at unknown locations.
 
   Other protection depends on the code family.
   The GNU PI family (parameterized by \(t\)) protects against arbitrary weight \(t\) qubit errors and approximately corrects spontaneous decay errors \cite{arxiv:1302.3247,doi:10.1103/PhysRevA.93.042340}.

codes/quantum/qubits/qubits_into_qubits.yml

@@ -47,7 +47,8 @@ protection: |
   \hyperref[topic:degeneracy]{Degenerate} codes admit undetectable Pauli errors (i.e., errors whose projection into the codespace is nonzero) of weight less than the code distance (i.e., the projection satisfies the \term{Knill-Laflamme conditions}).
 
   A quantum channel whose Kraus operators are Pauli strings is called a \textit{Pauli channel}, and such channels are typically more tractable than general, non-Pauli channels.
-  Relevant Pauli channels include dephasing noise and depolarizing noise (a.k.a. Werner-Holevo channel \cite{arxiv:quant-ph/0203003}), while relevant non-Pauli channels are amplitude damping, erasure (which maps all qubit states into a third state \(|e\rangle\) outside of the qubit Hilbert space), and biased erasure (in which case only the \(|1\rangle\) qubit state is mapped to \(|e\rangle\)).
+  Relevant Pauli channels include dephasing noise and depolarizing noise (a.k.a. Werner-Holevo channel \cite{arxiv:quant-ph/0203003}).
+  Relevant non-Pauli channels are amplitude damping (which requires protection against \(X+iY\) instead of \(X,Y\) individually), erasure (which maps all qubit states into a third state \(|e\rangle\) outside of the qubit Hilbert space), and biased erasure (in which case only the \(|1\rangle\) qubit state is mapped to \(|e\rangle\)).
   Noise can be correlated in space or in time, with the latter being an example of a non-Markovian phenomenon \cite{arxiv:2012.01894}.
 
   \subsection{Bounds on code parameters}