~

Author: valbert4

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

@@ -53,6 +53,7 @@ protection: |
 relations:
   parents:
     - code_id: q-ary_linear
+    - code_id: dual_over_rings
 
 
 # Begin Entry Meta Information

codes/classical/rings/dual/dual_over_rings.yml

@@ -10,18 +10,18 @@ logical: rings
 name: 'Dual linear code over \(R\)'
 
 description: |
-  For any linear code \(C\) over a ring \(R\), the dual code is the set of strings that are orthogonal to the codewords of \(C\) under a particular inner product.
-
-  The dual code over \(R\) is a linear code defined by
-  \begin{align}
-  C^\perp = \{ y\in R^{n} ~|~ x \cdot y=0 \forall x\in C\},
-  \end{align}
-  where the \textit{ordinary}, \textit{standard}, or \textit{Euclidean} inner product is \(x\cdot y = \sum_{i=1}^n x_i y_i\) for coordinates \(x_i,y_i\).
-
-  A code that is contained in its dual, \(C \subseteq C^\perp\), is called \textit{self-orthogonal over \(R\)} or \textit{weakly self-dual over \(R\)}.
-  A code that contains its dual, \(C^\perp \subseteq C\), is called \textit{dual-containing over \(R\)}.
-  A code that is equal to its dual, \(C^\perp = C\), is called self-dual over \(R\).
-  A code is dual-containing over \(R\) iff its dual is self-orthogonal over \(R\).
+  For any linear code \(C\) over a ring \(R\), the dual code is the set of strings that are orthogonal to the codewords of \(C\) under some inner product.
+
+# The dual code over \(R\) is a linear code defined by
+# \begin{align}
+# C^\perp = \{ y\in R^{n} ~|~ x \cdot y=0 \forall x\in C\},
+# \end{align}
+# where the \textit{ordinary}, \textit{standard}, or \textit{Euclidean} inner product is \(x\cdot y = \sum_{i=1}^n x_i y_i\) for coordinates \(x_i,y_i\).
+
+# A code that is contained in its dual, \(C \subseteq C^\perp\), is called \textit{self-orthogonal over \(R\)} or \textit{weakly self-dual over \(R\)}.
+# A code that contains its dual, \(C^\perp \subseteq C\), is called \textit{dual-containing over \(R\)}.
+# A code that is equal to its dual, \(C^\perp = C\), is called self-dual over \(R\).
+# A code is dual-containing over \(R\) iff its dual is self-orthogonal over \(R\).
 
 
 relations:

codes/classical/rings/over_zq/dual/dual_over_zq.yml

@@ -24,7 +24,8 @@ protection: |
 
 relations:
   parents:
-    - code_id: quaternary_over_z4
+    - code_id: q-ary_linear_over_zq
+    - code_id: dual_over_rings
 
 
 # Begin Entry Meta Information