~

Author: valbert4

codes/classical/bits/nonlinear/gray_map/originals/nordstrom_robinson.yml

@@ -34,9 +34,13 @@ relations:
       detail: 'The NR code is the smallest Kerdock code.'
     - code_id: preparata
       detail: 'The NR code is the smallest Preparata code.'
+    - code_id: orthogonal_array
+      detail: 'The NR code is an orthogonal array of strength \(5\) \cite[pg. 141]{preset:MacSlo}.'
   cousins:
     - code_id: octacode
       detail: 'The NR code is the image of the octacode under the \term{Gray map}  \cite{manual:{Forney Jr GD, Sloane NJ, Trott MD. The Nordstrom-Robinson code is the binary image of the octacode. In Coding and Quantization: DIMACS/IEEE workshop 1992 Oct 19 (pp. 19-26). Amer. Math. Soc..},doi:10.1142/3603}. The \((14, 64, 6)\) shortened NR code is the image of the heptacode under the \term{Gray map} \cite[Exam. 5]{doi:10.1109/TIT.2021.3114636}.'
+    - code_id: biorthogonal
+      detail: 'The NR code is the union of eight cosets of a linear \([16,5,8]\) code, i.e., the first-order Reed-Muller (biorthogonal) code \cite[pgs. 76 and 476]{preset:MacSlo}.'
     - code_id: extended_golay
       detail: 'The NR code can be constructed using the extended Golay code by first selecting a set of codewords satisfying certain conditions and then deleting specific coordinates \cite[pg. 73]{preset:MacSlo}.'
     - code_id: self_dual

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

@@ -30,6 +30,8 @@ relations:
   parents:
     - code_id: sloane_whitehead
   cousins:
+    - code_id: combinatorial_design
+      detail: 'Julin-Golay codes are constructed from the Steiner system \(S(5,6,12)\) arising from the extended \((12,132,4)\) code \cite[pgs. 70-72]{preset:MacSlo}.'
     - code_id: sphere_packing
       detail: 'Using \term{Construction A}, the Julin-Golay codes yield non-lattice sphere-packings that hold records in 9 and 11 dimensions.'
     - code_id: construction_a

codes/classical/bits/reed_muller/dual_hamming/hadamard.yml

@@ -35,6 +35,8 @@ relations:
   cousins:
     - code_id: long
       detail: 'The Hadamard code is a subcode of the long code and can be obtained by restricting the long-code construction to only linear functions.'
+    - code_id: binary_quad_residue
+      detail: 'For Hadamard matrices obtained from the Paley construction, the linear span of the resulting Hadamard codes yields quadratic-residue codes \cite[pg. 49]{preset:MacSlo}.'
     - code_id: reed_muller
       detail: 'The \([2^m,m+1,2^{m-1}]\) augmented Hadamard code is the first-order RM code (a.k.a. RM\((1,m)\)). The \([2^m-1,m,2^{m-1}]\) shortened Hadamard code is the simplex code (a.k.a. RM\(^*(1,m)\)). Rows of a Hadamard matrix forming a Prometheus orthonormal set (PONS) are codewords of a coset of RM\((1,m)\) in RM\((2,m)\) \cite{doi:10.1007/1-4020-2307-3_7}.'
     - code_id: simplex

codes/classical/bits/reed_muller/dual_hamming/repetition.yml

@@ -51,7 +51,7 @@ relations:
       detail: '\(q\)-ary repetition code reduce to repetition codes for \(q=2\).'
   cousins:
     - code_id: perfect_binary
-      detail: 'Repetition codes are perfect for odd \(n\).'
+      detail: 'Repetition codes are trivially perfect for odd \(n\) \cite[pg. 180]{preset:MacSlo}.'
     - code_id: quantum_repetition
       detail: 'A quantum repetition code can be thought of as a classical \([n,1,n]\) repetition code embedded in a quantum system.'
     - code_id: hamming

codes/classical/bits/reed_muller/dual_hamming/simplex.yml

@@ -38,13 +38,15 @@ features:
 
 relations:
   parents:
-    - code_id: binary_linear
-      detail: 'Linear binary codes cannot be constant weight, but can have nonzero codewords with constant weight. All such codes are equidistant, and Bonisoli''s theorem states that any equidistant linear binary code is a direct sum of simplex codes \cite{manual:{Bonisoli, Arrigo. "Every equidistant linear code is a sequence of dual Hamming codes." Ars Combinatoria 18 (1984): 181-186.}} (see also Refs. \cite{doi:10.1016/S0019-9958(63)80010-8,doi:10.1137/0114009}).'
+    - code_id: binary_cyclic
+      detail: 'Simplex codes can be realized as maximal-length feedback-shift-register codes, and are therefore cyclic \cite[pgs. 89 and 216]{preset:MacSlo}.'
     - code_id: combinatorial_design
       detail: 'Simplex codewords form a 2-design \cite[pg. 166]{preset:MacSlo}.'
     - code_id: q-ary_simplex
       detail: '\(q\)-ary simplex codes reduce to simplex codes for \(q=2\).'
   cousins:
+    - code_id: binary_linear
+      detail: 'Linear binary codes cannot be constant weight, but can have nonzero codewords with constant weight. All such codes are equidistant, and Bonisoli''s theorem states that any equidistant linear binary code is a direct sum of simplex codes \cite{manual:{Bonisoli, Arrigo. "Every equidistant linear code is a sequence of dual Hamming codes." Ars Combinatoria 18 (1984): 181-186.}} (see also Refs. \cite{doi:10.1016/S0019-9958(63)80010-8,doi:10.1137/0114009}).'
     - code_id: hamming
       detail: 'Hamming and simplex codes are dual to each other.'
     - code_id: dual