graph edits

Author: valbert4

codes/classical/analog/sphere_packing/lattice/root/eeight.yml

@@ -39,14 +39,14 @@ notes:
 
 relations:
   parents:
+    - code_id: self_dual_lattice
+      detail: 'The \(E_8\) Gosset lattice is even and unimodular.'
     - code_id: root
     - code_id: barnes_wall
     - code_id: construction_a
       detail: 'The \([8,4,4]\) extended Hamming code yields the \(E_8\) Gosset lattice via \term{Construction A} \cite[Exam. 10.5.2]{preset:EricZin}.'
     - code_id: construction_a4
       detail: 'The octacode yields the \(E_8\) Gosset lattice via \term{Construction \(A_4\)} \cite{doi:10.1007/3-540-57843-9_20,doi:10.1109/18.370138}\cite[Exam. 12.5.13]{doi:10.1017/CBO9780511807077}.'
-    - code_id: self_dual_lattice
-      detail: 'The \(E_8\) Gosset lattice is even and unimodular.'
     - code_id: univ_opt_analog
       detail: 'The \(E_8\) Gosset lattice is universally optimal \cite{arxiv:1902.05438}.'
   cousins:

codes/classical/homogeneous/symmetric/symmetric_space.yml

@@ -12,7 +12,7 @@ name: 'Symmetric-space code'
 description: |
   Encodes \(K\) states (codewords) into a symmetric space, which is a homogeneous space \(G/H\) with an additional property whose definition depends on whether the homogeneous space is continuous or Finite.
   
-  Continuous symmetric spaces are homogeneous spaces with an appropriately defined inversion operation. The two-sphere is a symmetric space, and its operation is inversion through the origin. This holds true in higher dimensions, yielding the \(D\)-dimensional spherical symmetric space family \(SO(D+1)/SO(D)\). 
+  Infinite symmetric spaces are homogeneous spaces with an appropriately defined inversion operation. The two-sphere is a symmetric space, and its operation is inversion through the origin. This holds true in higher dimensions, yielding the \(D\)-dimensional spherical symmetric space family \(SO(D+1)/SO(D)\). 
   Cartan classified the compact symmetric spaces whose \(G\) are simple real Lie groups \cite{doi:10.24033/asens.781,doi:10.1007/978-3-642-18245-7}.
   These spaces include spheres, projective spaces, and Grassmannians.
   Noncompact symmetric spaces include Euclidean and hyperbolic spaces.
@@ -32,7 +32,7 @@ notes:
 relations:
   parents:
     - code_id: homogeneous_space_classical
-      detail: 'Continuous symmetric spaces are homogeneous spaces with an appropriately defined inversion operation. Finite symmetric spaces are defined in coding theory as spaces admitting a generously transitive group action \cite[Def. 4.5]{arxiv:0909.4767}\cite[Sec. 3.4]{arxiv:1007.2905}. For multiplicity-free spaces such as symmetric spaces, the zonal spherical functions form an Abelian algebra, and the behavior of such functions can be used to obtain bounds on code parameters such as the Levenshtein bound \cite{manual:{V. I. Levenshtein, "On choosing polynomials to obtain bounds in packing problems." Proc. Seventh All-Union Conf. on Coding Theory and Information Transmission, Part II, Moscow, Vilnius. 1978.},manual:{V. I. Levenshtein, “On bounds for packings in n-dimensional Euclidean space”, Dokl. Akad. Nauk SSSR, 245:6 (1979), 1299–1303},manual:{V. I. Levenshtein. Bounds for packings of metric spaces and some of their applications. Problemy Kibernet, 40 (1983), 43-110.}}.'
+      detail: 'Infinite symmetric spaces are homogeneous spaces with an appropriately defined inversion operation. Finite symmetric spaces are defined in coding theory as spaces admitting a generously transitive group action \cite[Def. 4.5]{arxiv:0909.4767}\cite[Sec. 3.4]{arxiv:1007.2905}. For multiplicity-free spaces such as symmetric spaces, the zonal spherical functions form an Abelian algebra, and the behavior of such functions can be used to obtain bounds on code parameters such as the Levenshtein bound \cite{manual:{V. I. Levenshtein, "On choosing polynomials to obtain bounds in packing problems." Proc. Seventh All-Union Conf. on Coding Theory and Information Transmission, Part II, Moscow, Vilnius. 1978.},manual:{V. I. Levenshtein, “On bounds for packings in n-dimensional Euclidean space”, Dokl. Akad. Nauk SSSR, 245:6 (1979), 1299–1303},manual:{V. I. Levenshtein. Bounds for packings of metric spaces and some of their applications. Problemy Kibernet, 40 (1983), 43-110.}}.'
     #  Also in homogeneous_space_classical
   cousins:
     - code_id: points_into_lattices

codes/classical/matrices/sum-rank-metric/rank-metric/rank_metric.yml

@@ -54,7 +54,8 @@ relations:
     - code_id: sum_rank_metric
       detail: 'The sum-rank metric generalizes both the Hamming metric and the rank metric \cite{arxiv:1710.03109}.'
     - code_id: 2pt_homogeneous
-      detail: 'Matrices of dimension \(m\times n\) over \(\mathbb{F}_q\) are in one-to-one correspondence with bilinear forms, which form a finite two-point homogeneous space \cite{doi:10.1016/0097-3165(78)90015-8,doi:10.1007/978-94-010-9787-1_2,doi:10.1016/j.jcta.2010.05.006}\cite[Table 2]{arxiv:1007.2905}.'
+      detail: 'Matrices of dimension \(m\times n\) over \(\mathbb{F}_q\) form a finite two-point homogeneous space \cite{doi:10.1016/0097-3165(78)90015-8,doi:10.1007/978-94-010-9787-1_2,doi:10.1016/j.jcta.2010.05.006}\cite[Table 2]{arxiv:1007.2905}.'
+    #  are in one-to-one correspondence with bilinear forms, which
   cousins:
     - code_id: subspace
       detail: 'An \(m \times n\) rank-metric codeword \(A\) can be \textit{lifted} to a subspace codeword \((I | A)\) that generates an \(m\)-dimensional subspace \cite{arxiv:0711.0708}\cite[Def. 14.5.21]{preset:HKSprojective}.'

codes/classical/spherical/polytope/self_dual_polytope.yml

@@ -16,6 +16,9 @@ description: |
 relations:
   parents:
     - code_id: dual_polytope
+  cousins:
+    - code_id: self_dual_lattice
+    - code_id: self_dual
 
 
 # Begin Entry Meta Information