qudit RM

codes/classical/bits/reed_muller.yml

@@ -32,6 +32,7 @@ features:
     - 'Reed decoder with \(r+1\)-step majority decoding corrects \(\frac{1}{2}(2^{m-r}-1)\) errors \cite{doi:10.1109/irepgelc.1954.6499441} (see also Ch. 13 of Ref. \cite{preset:MacSlo}).'
     - 'Sequential code-reduction decoding \cite{doi:10.1109/TIT.1973.1055023}.'
     - 'First-order (\(r=1\)) RM codes admit specialized decoders \cite{manual:{E.C. Posner, \emph{Combinatorial Structures in Planetary Reconnaissance} in Error Correcting Codes, ed. H.B. Mann, Wiley, NY 1968.}}.'
+    - 'Matrix factorization is equivalent to decoding an RM\((n,n-3)\) code \cite{doi:10.1137/0209059}; see \cite{arxiv:1606.01904}.'
 
 
 realizations:

codes/quantum/qubits/rm/quantum_reed_muller.yml

@@ -30,7 +30,10 @@ features:
 relations:
   parents:
     - code_id: qubit_css
+    - code_id: qudit_reed_muller
+      detail: 'Prime-qudit RM codes reduce to quantum RM codes when \(q=p=2\).'
     - code_id: galois_reed_muller
+      detail: 'Galois-qudit RM codes reduce to quantum RM codes when \(q=2\).'
   cousins:
     - code_id: reed_muller
     - code_id: quantum_convolutional

codes/quantum/qubits/small_distance/small/stab_15_1_3.yml

@@ -8,7 +8,7 @@ physical: qubits
 logical: qubits
 name: '\([[15,1,3]]\) quantum Reed-Muller code'
 short_name: '\([[15,1,3]]\)'
-introduced: '\cite{arxiv:quant-ph/9610011,arxiv:quant-ph/0403025}'
+introduced: '\cite{arxiv:quant-ph/9610011,arXiv:quant-ph/9608026,arxiv:quant-ph/0403025}'
 # Middle ref cited in https://arxiv.org/pdf/1709.08658.pdf
 
 description: |

codes/quantum/qudits/polynomial.yml

@@ -7,13 +7,15 @@ code_id: polynomial
 physical: qudits
 logical: qudits
 
-name: 'Quantum Reed-Solomon code'
-short_name: 'Quantum RS'
+name: 'Prime-qudit quantum RS code'
 introduced: '\cite{arxiv:quant-ph/9910059}'
 # doi:10.1103/PhysRevA.77.012308?
 
+alternative_names:
+  - 'Prime-qudit polynomial code (QPyC)'
+
 description: |
-  Also called \textit{prime-qudit polynomial code (QPyC)}. Prime-qudit CSS code constructed using two Reed-Solomon codes.
+  Prime-qudit CSS code constructed using two Reed-Solomon codes.
 
   The original construction \cite{arxiv:quant-ph/9910059} was for a qubit code (\(p=2\)) by using a basis for a larger Galois field over \(GF(2)\), yielding an \([[kN,k(N-2K),K+1]]\) qubit code from a \([N,K,\delta]_{GF(2^k)}\) RS code with \(N=2^k-1\) and \(K=N-\delta+1\).
 
@@ -29,7 +31,7 @@ description: |
   \end{align}
 
 features:
-  magic_scaling_exponent: 'Triorthogonal quantum RS codes achieve a magic-state distillation scaling exponent \(\gamma\) that is arbitrarily close to zero \cite{arxiv:1811.08461}.'
+  magic_scaling_exponent: 'Triorthogonal prime-qudit RS codes achieve a magic-state distillation scaling exponent \(\gamma\) that is arbitrarily close to zero \cite{arxiv:1811.08461}.'
 
 relations:
   parents:
@@ -39,7 +41,7 @@ relations:
   cousins:
     - code_id: quantum_triorthogonal
       detail: 'Triorthogonality can be generalized to qudit codes.
-      Quantum RS codes achieve a magic-state distillation scaling exponent \(\gamma\) that is arbitrarily close to zero \cite{arxiv:1811.08461}.'
+      Prime-qudit RS codes achieve a magic-state distillation scaling exponent \(\gamma\) that is arbitrarily close to zero \cite{arxiv:1811.08461}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qudits/small/qudit_reed_muller.yml

@@ -0,0 +1,34 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: qudit_reed_muller
+physical: qudits
+logical: qudits
+
+name: 'Prime-qudit RM code'
+introduced: '\cite{arXiv:quant-ph/0502001,arxiv:1205.3104}'
+
+description: |
+  Modular-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes via the modular-qudit CSS construction.
+  An odd-prime-qudit CSS code family constructed from first-order punctured RM codes transversally implements a diagonal gate at any level of the \hyperref[topic:qudit-clifford-hierarchy]{qudit Clifford hierarchy} \cite{arxiv:1205.3104}.
+
+features:
+  magic_scaling_exponent: 'An odd-prime-qudit CSS code family constructed from first-order punctured RM codes can be used for qudit magic-state distillation \cite{arxiv:1205.3104}.'
+
+  transversal_gates: 'An odd-prime-qudit CSS code family constructed from first-order punctured RM codes transversally implements a diagonal gate at any level of the \hyperref[topic:qudit-clifford-hierarchy]{qudit Clifford hierarchy} \cite{arxiv:1205.3104}.'
+
+relations:
+  parents:
+    - code_id: qudit_css
+    - code_id: galois_reed_muller
+      detail: 'Galois-qudit RM codes reduce to prime-qudit RM codes when \(q\) is prime.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-03-01'