gray cousins

codes/classical/bits/nonlinear/gray.yml

@@ -16,8 +16,9 @@ description: |
   A simple example is the case \(n=2\), also known as the \textit{Gray map}, which produces the ordering \(0\to 00\), \(1\to 01\), \(2\to 11\), and \(3\to 10\).
   The Gray map differs in the last two numbers from the usual binary expansion of the natural numbers, which maps \(0\to 00\), \(1\to 01\), \(2\to 10\), and \(3\to 11\).
 
-  Layout out the Gray-map output strings counterclockwise on the corners of a 1D square, gray codes have been generalized such that nearest-neighbor strings differ by only one digit when the strings are arranged in higher-dimensional hypercubes \cite{doi:10.1002/j.1538-7305.1958.tb03887.x}.
-  Further generalizations called \textit{combinatorial Gray codes} \cite{doi:10.1137/0209013} refer to methods to generate organize combinatorial objects such that successive objects differ in some particular way. Particular \(q\)-ary extensions \cite{doi:10.1109/TSMCB.2012.2210706} of Gray codes may be useful in digital imaging and signal processing.
+  Gray codes have been generalized such that nearest-neighbor strings differ by only one digit when the strings are arranged in higher-dimensional hypercubes \cite{doi:10.1002/j.1538-7305.1958.tb03887.x}.
+  Further generalizations called \textit{combinatorial Gray codes} \cite{doi:10.1137/0209013} refer to methods to organize combinatorial objects such that successive objects differ in some particular way.
+  Particular \(q\)-ary extensions \cite{doi:10.1109/TSMCB.2012.2210706} of Gray codes may be useful in digital imaging and signal processing.
 
 features:
   encoders:

codes/classical/bits/nonlinear/kerdock.yml

@@ -41,6 +41,8 @@ relations:
   cousins:
     - code_id: quaternary_over_z4
       detail: 'Kerdock codes can be seen, via the Gray map, as extended linear cyclic codes over \(\mathbb{Z}_4\), where the appended bit is a zero sum check \cite{doi:10.1109/18.312154,arxiv:math/0207208}.'
+    - code_id: gray
+      detail: 'Kerdock codes can be seen, via the Gray map, as extended linear cyclic codes over \(\mathbb{Z}_4\), where the appended bit is a zero sum check \cite{doi:10.1109/18.312154,arxiv:math/0207208}.'
     - code_id: reed_muller
       detail: 'Kerdock code is a subcode of a second-order RM Code \cite[pg. 457]{preset:MacSlo}.
       It consists of a number of cosets of RM\((2,m)\) created by quotienting with RM\((1,m)\).'

codes/classical/bits/nonlinear/preparata.yml

@@ -40,6 +40,8 @@ relations:
   cousins:
     - code_id: quaternary_over_z4
       detail: 'Preparata codes can be seen, via the Gray map, as linear codes over \(\mathbb{Z}_4\) \cite{doi:10.1109/18.312154,arxiv:math/0207208}.'
+    - code_id: gray
+      detail: 'Preparata codes can be seen, via the Gray map, as linear codes over \(\mathbb{Z}_4\) \cite{doi:10.1109/18.312154,arxiv:math/0207208}.'
     - code_id: quasi_perfect
       detail: 'Punctured Preparata codes are quasi-perfect \cite[pg. 475]{preset:MacSlo}.'
     - code_id: reed_muller

codes/classical/bits/nonlinear/sphere_packing/julin12.yml

@@ -36,6 +36,8 @@ relations:
       detail: 'Using Construction \(A\), the Julin-Golay codes yield non-lattice sphere-packings that hold records in 9 and 11 dimensions.'
     - code_id: q-ary_over_zq
       detail: 'Julin codes can be obtained from simple nonlinear codes over \(\mathbb{Z}_4\) using the Gray map \cite{doi:10.1007/BF01388558}.'
+    - code_id: gray
+      detail: 'Julin codes can be obtained from simple nonlinear codes over \(\mathbb{Z}_4\) using the Gray map \cite{doi:10.1007/BF01388558}.'
 
 
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codes/classical/bits/reed_muller.yml

@@ -57,6 +57,8 @@ relations:
       detail: 'RM\(^*(r,m)\) codes are equivalent to subcodes of BCH codes of designed distance \(2^{m-r}-1\) while RM\((r,m)\) are subcodes of extended BCH codes of the same designed distance \cite[Ch. 13]{preset:MacSlo}.'
     - code_id: quaternary_over_z4
       detail: 'RM codes are images of linear quaternary codes over \(\mathbb{Z}_4\) under the Gray map \cite[Sec. 6.3]{preset:HKSrings}.'
+    - code_id: gray
+      detail: 'RM codes are images of linear quaternary codes over \(\mathbb{Z}_4\) under the Gray map \cite[Sec. 6.3]{preset:HKSrings}.'
     - code_id: dual
       detail: 'The codes RM\((r,m)\) and RM\((m-r-1,m)\) are dual to each other.'
     - code_id: binary_duadic

codes/classical/rings/pentacode.yml

@@ -21,6 +21,8 @@ relations:
   cousins:
     - code_id: best
       detail: 'Codewords of the Best code can be obtained by applying the Gray map to the pentacode \cite[Sec. 2]{doi:10.1007/BF01388558}.'
+    - code_id: gray
+      detail: 'Codewords of the Best code can be obtained by applying the Gray map to the pentacode \cite[Sec. 2]{doi:10.1007/BF01388558}.'
 
 
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