~

codes/classical/bits/nonlinear/constantin_rao.yml

@@ -14,7 +14,7 @@ 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}.
+  A CR code for group \(G\) and fixed group element \(g\) consists of all binary strings \(c=c_1c_2\cdots c_n\) that satisfy \(\sum_i c_i g_i = g\) 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.}}.
@@ -25,8 +25,8 @@ protection: |
 
 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 Refs. \cite{doi:10.1016/S0019-9958(80)90082-0,doi:10.1109/18.651063}.'
+    CR codes for particular groups have higher rates than distance-one 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 Refs. \cite{doi:10.1016/S0019-9958(80)90082-0,doi:10.1109/18.651063}.
 
 
 relations:

codes/classical/bits/nonlinear/vt_single_deletion.yml

@@ -19,6 +19,7 @@ description: |-
   C_{n,a}=\left\{x\in\{0,1\}^n: \sum_{i=1}^n i~x_i = a\mod (n+1) \right\}.
   \end{align}
 
+  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}.
   VT codes can be generalized to the \(q\)-ary case \cite{arxiv:1708.04071,doi:10.1109/TIT.1984.1056962,arxiv:1906.07887}.
 
 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 \(a\).'
@@ -34,10 +35,7 @@ relations:
     - 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).}}, VT codes can be modified to yield slightly longer linear codes \cite{arxiv:math/0207197}.'
-
+
 
 # Begin Entry Meta Information
 _meta:

codes/classical/properties/block/universally_optimal/t-designs.yml

@@ -20,7 +20,7 @@ description: |
   \begin{align}
     \int_{X}dxp(x)={\textstyle \frac{1}{|D|}}\sum_{x\in D}p(x)~,
   \end{align}
-  where the integral is over \(X\) (given some measure \(d x\)), while the sum is over the design \(D\).
+  where the integral is over \(X\) (given some measure \(d x\)), while the sum is over the design \(D\subset X\).
   A \textit{weighted design} is a design for which each term \(p(x)\) in the above sum must be multiplied by a weight \(w(x)\) in order to be equal to the left-hand side.
   The most well-known examples of designs are (exact) quadrature/cubature formulas for integration over the reals \cite{manual:{Stroud, Arthur H. Approximate calculation of multiple integrals. Prentice Hall, 1971.},doi:10.1017/S0962492900002701,doi:10.1016/S0885-064X(03)00011-6,doi:10.18434/M3167}, \(X = \mathbb{R}^D\) (with appropriate measure); these tend to be weighted designs.