reed_solomon_4

Author: valbert4

codes/classical/analog/sphere_packing/lattice/dual/eeight.yml

@@ -52,8 +52,6 @@ relations:
   cousins:
     - code_id: hamming844
       detail: 'The \([8,4,4]\) extended Hamming code yields the \(E_8\) Gosset lattice via \term{Construction A} \cite[Exam. 10.5.2]{preset:EricZin}\cite[pg. 138]{doi:10.1007/978-1-4757-6568-7}.'
-    - code_id: q-ary_hamming
-      detail: 'The \([4,2,3]_3\) ternary Hamming code can be used to obtain the \(E_8\) Gosset lattice \cite[Exam. 10.5.5]{preset:EricZin}.'
     - code_id: repetition
       detail: 'The \([8,1,8]\) repetition code can be used to obtain the \(E_8\) Gosset lattice \cite[Exam. 10.7.1]{preset:EricZin}.'
     - code_id: sharp_config

codes/classical/q-ary_digits/ag/reed_solomon/rs/reed_solomon.yml

@@ -102,10 +102,6 @@ relations:
       detail: 'RS codes can be used to construct LTCs encoding \(k\) bits with length \(k \text{polylog}(k)\) and query complexity \(\text{polylog}(k)\) \cite{doi:10.1137/050646445}.'
     - code_id: pir
       detail: 'RS codes can be used to construct PIR codes \cite{doi:10.1109/ISIT45174.2021.9517900}.'
-    - code_id: hamming844
-      detail: 'The \([4,2,3]_4\) RS code is the smallest example of a quaternary quadratic-residue code and can be mapped to the \([8,4,4]\) extended Hamming code \cite[Sec. 2.4.2]{doi:10.1007/3-540-30731-1} by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) \cite{doi:10.1006/ffta.2001.0333}.'
-    - code_id: q-ary_quad_residue
-      detail: 'The \([4,2,3]_4\) RS code is the smallest example of a quaternary quadratic-residue code and can be mapped to the \([8,4,4]\) extended Hamming code \cite[Sec. 2.4.2]{doi:10.1007/3-540-30731-1} by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) \cite{doi:10.1006/ffta.2001.0333}.'
 
 
 

codes/classical/q-ary_digits/easy/q-ary_hamming.yml

@@ -34,8 +34,6 @@ relations:
       detail: 'Hamming codes are equivalent to cyclic codes when \(q\) and \(r\) are relatively prime (\cite{preset:MacSlo}, pg. 194).'
     - code_id: bch
       detail: 'Some narrow sense BCH codes of length \(n=(q^r-1)/(q-1)\) such that \(\text{gcd}(r,q-1)=1\) are \(q\)-ary Hamming codes (\cite{doi:10.1017/CBO9780511807077}, Thm. 5.1.4).'
-    - code_id: mds
-      detail: 'The \([4,2,3]_3\) Hamming code is a unique MDS code \cite{manual:{Taussky, Olga, and John Todd. "Covering theorems for groups." Bulletin of the American Mathematical Society. Vol. 54. No. 3. 201 CHARLES ST, PROVIDENCE, RI 02940-2213: AMER MATHEMATICAL SOC, 1948.},doi:10.1112/jlms/s1-44.1.60}.'
 
 
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codes/classical/q-ary_digits/easy/reed_solomon_4.yml

@@ -0,0 +1,49 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: reed_solomon_4
+physical: q-ary_digits
+logical: q-ary_digits
+
+name: '\([4,2,3]_4\) RS\(_4\) code'
+short_name: 'RS\(_4\)'
+
+alternative_names:
+  - '\(XQ(\mathbb{F}_4,3)\)'
+
+description: |
+  A Type II Euclidean self-dual RS code that is the smallest quaternary extended QR code \cite[Sec. 2.4.2]{doi:10.1007/3-540-30731-1}.
+
+  A generator matrix for the code is
+  \begin{align}
+    \begin{pmatrix}
+    1 & 1 & 1 & 1 \\
+    0 & 1 & \omega & \omega^2
+    \end{pmatrix}~,
+  \end{align}
+  where \(\mathbb{F}_4 = \{0,1,\omega, \bar{\omega}\}\) is the \hyperref[topic:finite-fields]{quaternary Galois field}.
+
+  The automorphism group of the code is \(3.S_4\) \cite[Sec. 2.4.2]{doi:10.1007/3-540-30731-1}.
+
+
+relations:
+  parents:
+    - code_id: self_dual
+      detail: 'The RS\(_4\) is the smallest Type II Euclidean self-dual code \cite[Sec. 2.4.2]{doi:10.1007/3-540-30731-1}.'
+    - code_id: reed_solomon
+    - code_id: small_distance
+  cousins:
+    - code_id: q-ary_quad_residue
+      detail: 'The RS\(_4\) code is the smallest quaternary extended QR code \cite[Sec. 2.4.2]{doi:10.1007/3-540-30731-1}.'
+    - code_id: hamming844
+      detail: 'The RS\(_4\) code can be mapped to the \([8,4,4]\) extended Hamming code \cite[Sec. 2.4.2]{doi:10.1007/3-540-30731-1} by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) \cite{doi:10.1006/ffta.2001.0333}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2026-01-03'

codes/classical/q-ary_digits/easy/tetracode.yml

@@ -34,6 +34,7 @@ relations:
     - code_id: q-ary_hamming
       detail: 'The tetracode is equivalent to the \(r=2\) \(3\)-ary Hamming code.'
     - code_id: mds
+      detail: 'The tetracode is a unique MDS code \cite{manual:{Taussky, Olga, and John Todd. "Covering theorems for groups." Bulletin of the American Mathematical Society. Vol. 54. No. 3. 201 CHARLES ST, PROVIDENCE, RI 02940-2213: AMER MATHEMATICAL SOC, 1948.},doi:10.1112/jlms/s1-44.1.60}.'
     - code_id: extended_reed_solomon
       detail: 'The tetracode is an extended RS code \cite[pg. 81]{doi:10.1007/978-1-4757-6568-7}.'
     - code_id: lexicographic
@@ -42,6 +43,8 @@ relations:
     - code_id: ternary_golay
       detail: 'Extended ternary Golay codewords can be obtained from tetracodewords \cite{doi:10.1007/978-1-4757-6568-7}.
       The tetracode can be used to decode the extended ternary Golay code \cite{doi:10.1109/TIT.1986.1057197}.'
+    - code_id: eeight
+      detail: 'The \([4,2,3]_3\) tetracode can be used to obtain the \(E_8\) Gosset lattice \cite[Exam. 10.5.5]{preset:EricZin}.'
 
 
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