gbch, srivastava, generalized_srivastava

codes/classical/q-ary_digits/ag/residueAG/goppa.yml

@@ -39,6 +39,8 @@ relations:
     - code_id: residue
     - code_id: cartier
       detail: 'Goppa codes are Cartier codes from a curve of genus zero \cite{arxiv:1206.4728}.'
+    - code_id: alternant
+      detail: 'Goppa codes are a special case of alternant codes \cite[Ch. 12]{preset:MacSlo}.'
   cousins:
     - code_id: generalized_reed_solomon
       detail: 'Goppa codes are \(GF(q)\)-\hyperref[topic:finite-fields]{subfield} subcode of the dual of the GRS code over \(GF(q^m)\) with evaluation points \(\alpha_i\) and factors \(v_i=G(\alpha_i)^{-1}\) (\cite{doi:10.1017/CBO9780511807077}, pg. 523; \cite{preset:HPAlgCodes}).' #alternant

codes/classical/q-ary_digits/ag/rs/alternant.yml → codes/classical/q-ary_digits/alternant/alternant.yml

@@ -8,9 +8,9 @@ physical: q-ary_digits
 logical: q-ary_digits
 
 name: 'Alternant code'
-introduced: '\cite{doi:10.1016/S0019-9958(74)80005-7}'
+introduced: '\cite{doi:10.1016/S0019-9958(72)80007-X,doi:10.1016/S0019-9958(74)80005-7,doi:10.1016/S0019-9958(75)90099-6,doi:10.1109/TIT.1975.1055435}'
 
-description: 'Given a length-\(n\) GRS code \(C\) over \(GF(q^m)\), an alternant code is the \(GF(q)\)-\hyperref[topic:finite-fields]{subfield} subcode of the dual of \(C\).'
+description: 'Given a length-\(n\) GRS code \(C\) over \(GF(q^m)\), an alternant code is the \(GF(q)\)-\hyperref[topic:finite-fields]{subfield} subcode of the dual of \(C\); see \cite[Ch. 12]{preset:MacSlo}. Its parity-check matrix is an alternant matrix.'
 
 # More precisely, an \([n, k \geq n-tm, d \geq t + 1]\) code with parity check matrix with \(n\) columns and \(t\) rows of the form:
 # \begin{align}
@@ -31,6 +31,7 @@ description: 'Given a length-\(n\) GRS code \(C\) over \(GF(q^m)\), an alternant
 features:
   decoders:
     - 'Variation of the Berlekamp-Welch algorithm \cite{doi:10.1109/TIT.1977.1055730}.'
+    - 'Euclidean algorithm; see \cite[Ch. 12]{preset:MacSlo} for more details.'
     - 'Guruswami-Sudan list decoder \cite{doi:10.1109/18.782097,doi:10.1109/SFCS.1998.743426}.'
 
 notes:
@@ -41,7 +42,7 @@ relations:
     - code_id: q-ary_linear
   cousins:
     - code_id: generalized_reed_solomon
-      detail: 'Alternant codes are \hyperref[topic:finite-fields]{subfield} subcodes of GRS codes.'
+      detail: 'Alternant codes are \hyperref[topic:finite-fields]{subfield} subcodes of GRS codes \cite{doi:10.1109/TIT.1975.1055435}.'
 
 #Classic (L,G)-Goppa codes are a subclass of alternant codes, and they are also cousins of AG codes. Alternant codes as such don't qualify as cousins of AG codes. They are in fact \hyperref[topic:finite-fields]{subfield} subcodes of GRS codes and as such are cousins of BCH codes and children of GRS codes.
 #A \hyperref[topic:finite-fields]{subfield} subcode is a subset of the code formed of the vectors all of whose coordinates are in a \hyperref[topic:finite-fields]{subfield} (or even just a subset, they have considered subset subcodes, where this subalphabet does not have to be a field, e.g., 49 -> 46).

codes/classical/q-ary_digits/alternant/gbch.yml

@@ -0,0 +1,31 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: gbch
+physical: q-ary_digits
+logical: q-ary_digits
+
+name: 'Chien-Choy generalized BCH (GBCH) code'
+short_name: 'GBCH'
+introduced: '\cite{doi:10.1109/TIT.1975.1055336}'
+
+description: |
+  An \([n,k\geq n-rm, d\geq r+1]_q\) alternant code defined using two polynomials \(P(x),G(x)\) that are relatively prime to \(x^n-1\), with \(\deg P \leq n-1\) and \(r = \deg G \leq n-1\).
+
+  See \cite[Ch. 12]{preset:MacSlo} for the parity-check matrix.
+
+
+relations:
+  parents:
+    - code_id: alternant
+      detail: 'GBCH codes are a special case of alternant codes \cite[Ch. 12]{preset:MacSlo}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-08-18'

codes/classical/q-ary_digits/alternant/generalized_srivastava.yml

@@ -0,0 +1,37 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: generalized_srivastava
+physical: q-ary_digits
+logical: q-ary_digits
+
+name: 'Generalized Srivastava code'
+introduced: '\cite{doi:10.1109/TIT.1972.1054760}'
+
+description: |
+  An \([n,k \geq n-mst,d \geq st+1 ]_q) alternant code defined for \(n+s\) distinct elements \(\alpha_1,\alpha_2,\cdots,\alpha_n,w_1,w_2,\cdots,w_s\) and \(n\) nonzero elements \(z_1,z_2,\cdots,z_n\) of \(GF(q^m)\).
+
+  The code's parity-check matrix is \cite[Ch. 12]{preset:MacSlo}
+  \begin{align}
+    H=\begin{pmatrix}\frac{z_{1}}{\alpha_{1}-w_{1}} & \frac{z_{2}}{\alpha_{2}-w_{1}} & \cdots & \frac{z_{n}}{\alpha_{n}-w_{1}}\\
+    \frac{z_{1}}{\left(\alpha_{1}-w_{2}\right)^{2}} & \frac{z_{2}}{\left(\alpha_{1}-w_{2}\right)^{2}} & \cdots & \frac{z_{n}}{\left(\alpha_{n}-w_{2}\right)^{2}}\\
+    \vdots & \vdots & \ddots & \vdots\\
+    \frac{z_{1}}{\left(\alpha_{1}-w_{s}\right)^{t}} & \frac{z_{2}}{\left(\alpha_{2}-w_{s}\right)^{t}} & \cdots & \frac{z_{n}}{\left(\alpha_{n}-w_{s}\right)^{t}}
+    \end{pmatrix}~.
+  \end{align}
+
+
+relations:
+  parents:
+    - code_id: alternant
+      detail: 'Generalized Srivastava codes are a special case of alternant codes \cite[Ch. 12]{preset:MacSlo}.'
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-08-18'

codes/classical/q-ary_digits/group/cyclic/q-ary_bch.yml → codes/classical/q-ary_digits/alternant/q-ary_bch.yml

@@ -38,17 +38,20 @@ realizations:
 
 relations:
   parents:
+    - code_id: gbch
+      detail: '\(q\)-ary BCH codes are a special case of GBCH codes \cite[Ch. 12]{preset:MacSlo}.'
     - code_id: q-ary_cyclic
     - code_id: twisted_bch
       detail: 'Twisted BCH codes are obtained from \(q\)-ary BCH codes via various lengthening and extension procedures.'
   cousins:
     - code_id: reed_solomon
       detail: 'BCH codes are \hyperref[topic:finite-fields]{subfield} subcodes of RS codes.'
     - code_id: goppa
-      detail: 'Narrow-sense BCH codes are Goppa codes with \(L=\{1,\alpha^{-1},\cdots,\alpha^{1-n}\}\) and \(G(x)=x^{\delta-1}\) (\cite{doi:10.1017/CBO9780511807077}, pg. 522).'
+      detail: 'Narrow-sense BCH codes are Goppa codes with \(L=\{1,\alpha^{-1},\cdots,\alpha^{1-n}\}\) and \(G(x)=x^{\delta-1}\) \cite[pg. 522]{doi:10.1017/CBO9780511807077}.'
     - code_id: q-ary_ltc
       detail: 'Duals of BCH codes are locally testable \cite{doi:10.1109/SFCS.2005.16}.'
 
+
 # Begin Entry Meta Information
 _meta:
   # Change log - most recent first

codes/classical/q-ary_digits/alternant/srivastava.yml

@@ -0,0 +1,45 @@
+#######################################################
+## This is a code entry in the error correction zoo. ##
+##       https://github.com/errorcorrectionzoo       ##
+#######################################################
+
+code_id: srivastava
+physical: q-ary_digits
+logical: q-ary_digits
+
+name: 'Srivastava code'
+introduced: '\cite{doi:10.1142/9407,doi:10.1109/TIT.1972.1054760}'
+
+description: |
+  A special case of a generalized Srivastava code for \(z_j = \alpha_j^{\mu}\) for some \(\mu\) and \(t=1\).
+
+  The code's parity-check matrix is
+  \begin{align}
+    H=\begin{pmatrix}\frac{\alpha_{1}^{\mu}}{\alpha_{1}-w_{1}} & \frac{\alpha_{2}^{\mu}}{\alpha_{2}-w_{1}} & \cdots & \frac{\alpha_{n}^{\mu}}{\alpha_{n}-w_{1}}\\
+    \frac{\alpha_{1}^{\mu}}{\alpha_{1}-w_{2}} & \frac{\alpha_{2}^{\mu}}{\alpha_{1}-w_{2}} & \cdots & \frac{\alpha_{n}^{\mu}}{\alpha_{n}-w_{2}}\\
+    \vdots & \vdots & \ddots & \vdots\\
+    \frac{\alpha_{1}^{\mu}}{\alpha_{1}-w_{s}} & \frac{\alpha_{2}^{\mu}}{\alpha_{2}-w_{s}} & \cdots & \frac{\alpha_{n}^{\mu}}{\alpha_{n}-w_{s}}
+    \end{pmatrix}~.
+  \end{align}
+
+protection: |
+  Dimension and minimum distance are found in Refs. \cite{doi:10.1109/TIT.1972.1054760,doi:10.1016/S0019-9958(72)80007-X}.
+
+
+relations:
+  parents:
+    - code_id: generalized_srivastava
+      detail: 'A Srivastava code is a special case of a generalized Srivastava code for \(z_j = \alpha_j^{\mu}\) for some \(\mu\) and \(t=1\).'
+    - code_id: goppa
+      detail: 'Generalized Srivastava codes are a special case of Goppa codes \cite[Ch. 12]{preset:MacSlo}.'
+    - code_id: gbch
+      detail: 'Generalized Srivastava codes are a special case of GBCH codes \cite[Ch. 12]{preset:MacSlo}.'
+
+
+
+# Begin Entry Meta Information
+_meta:
+  # Change log - most recent first
+  changelog:
+    - user_id: VictorVAlbert
+      date: '2024-08-18'