remove redundant primary children, lint

valbert4 · valbert4 · commit 209835c811ec · 2026-05-23T18:13:40.000-04:00
diff --git a/codelists/features/quantum/list_transversal.yml b/codelists/features/quantum/list_transversal.yml
@@ -2,12 +2,14 @@
 list_id: 'quantum_transversal'
 
 title: |
-  Quantum codes with transversal gates
+  Quantum codes with transversal or permutation-based gates
 
 
 intro: |
-  Here is a list of all quantum codes that admit transversal gates. Applicable to codes living in a tensor-product space,
-  such gates can be written as a tensor product of unitary operations, with each operation acting on its corresponding subsystem.
+  Here is a list of all quantum codes that admit transversal or permutation-based gates. Applicable to codes living in a tensor-product space,
+  transversal gates can be written as a tensor product of unitary operations, with each operation acting on its corresponding subsystem.
+  Permutation gates permute subsystems.
+
 
 display:
   style: table
diff --git a/codes/classical/bits/constant_weight/constant_weight.yml b/codes/classical/bits/constant_weight/constant_weight.yml
@@ -38,9 +38,9 @@ relations:
     - code_id: bits_into_bits
     - code_id: q-ary_constant_weight
       detail: 'The set of all weight-\(w\) binary strings of length \(n\) forms the \textit{Johnson space} \(J(n,w)\), a finite two-point homogeneous space \(G/H\) with \(G = S_n\) and \(H = S_w \times S_{n-w}\) \cite{preset:Delsarte73,doi:10.1016/0097-3165(76)90017-0,doi:10.1137/0134012,doi:10.1007/BF00053379}\cite[Sec. 4.2.1]{arxiv:0909.4767}\cite[Table 2]{arxiv:1007.2905}. This is a special case of the nonbinary Johnson space for \(q=2\).'
+  cousins:
     - code_id: 2pt_homogeneous
       detail: 'The set of all weight-\(w\) binary strings of length \(n\) forms the \textit{Johnson space} \(J(n,w)\), a finite two-point homogeneous space \(G/H\) with \(G = S_n\) and \(H = S_w \times S_{n-w}\) \cite{preset:Delsarte73,doi:10.1016/0097-3165(76)90017-0,doi:10.1137/0134012,doi:10.1007/BF00053379}\cite[Sec. 4.2.1]{arxiv:0909.4767}\cite[Table 2]{arxiv:1007.2905}.'
-  cousins:
     - code_id: finite_grassmann
       detail: 'Codewords of length \(n\) and weight \(w\) are in one-to-one correspondence with subsets of \(n\) objects with \(w\) elements. The \(q\)-Johnson spaces generalize this notion to subspaces and reduce to Johnson spaces at \(q=1\). In other words, \((q=2)\)-Johnson space is not the same as (binary) Johnson space since the former indexes subspaces, while the latter indexes subsets.'
     - code_id: delsarte_optimal
diff --git a/codes/classical/bits/cyclic/quad_residue/extended_golay.yml b/codes/classical/bits/cyclic/quad_residue/extended_golay.yml
@@ -41,13 +41,13 @@ relations:
       detail: 'The extended Golay code is a sharp configuration \cite[Table 12.1]{preset:HKSbounds}.'
     - code_id: karlin
       detail: 'The extended Golay code is equivalent to the Karlin double circulant code for \(m=11\) \cite[Ch. 16]{preset:MacSlo}.'
-    - code_id: orthogonal_array
-      detail: 'The extended Golay code is an orthogonal array of strength 7 \cite[Exam. 1]{doi:10.1109/18.720545}.'
     - code_id: quasi_group
       detail: 'The extended Golay code is a quasi group-algebra code for various groups \cite{doi:10.1016/0097-3165(90)90069-9,arxiv:1604.07863,arxiv:1912.09167}.'
     - code_id: lexicographic
       detail: 'The extended Golay code is a lexicode \cite{manual:{M. J. T. Guy, unpublished},doi:10.1109/TIT.1986.1057187}\cite[pg. 327]{preset:MacSlo}.'
   cousins:
+    - code_id: orthogonal_array
+      detail: 'The extended Golay code is an orthogonal array of strength 7 \cite[Exam. 1]{doi:10.1109/18.720545}.'
     - code_id: self_dual
       detail: 'The extended Golay code is the unique \([24,12,8]\) code, and in particular the unique self-dual doubly even code with those parameters \cite{doi:10.1016/0012-365X(75)90047-3}\cite[Rem. 4.3.11]{preset:HKSselfdual}.'
     - code_id: biorthogonal
diff --git a/codes/classical/bits/reed_muller/dual_hamming/hadamard.yml b/codes/classical/bits/reed_muller/dual_hamming/hadamard.yml
@@ -30,9 +30,9 @@ relations:
       detail: 'Each nonzero Hadamard codeword has length \(2^m\) and Hamming weight of \(2^{m-1}\), making this code balanced.'
     - code_id: q-ary_lcc
       detail: 'Hadamard codes are two-query LDCs and LCCs \cite{doi:10.1561/0400000030,preset:Gopi18}.'
+  cousins:
     - code_id: locally_recoverable
       detail: 'The Hadamard code is an LRC with \(r=3\) \cite{arxiv:2311.08653}.'
-  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
diff --git a/codes/classical/bits/reed_muller/dual_hamming/repetition.yml b/codes/classical/bits/reed_muller/dual_hamming/repetition.yml
@@ -46,11 +46,11 @@ relations:
     - code_id: nearly_perfect
     - code_id: binary_cyclic
       detail: 'The repetition code is cyclic with generator polynomial \(1+x+\cdots+x^{n-1}\).'
-    - code_id: mds
-      detail: 'Binary repetition codes are trivial MDS codes \cite[Thm. 12.3.1]{preset:HKSbounds}\cite[Sec. 3.3.2]{preset:HKSclass}.'
     - code_id: q-ary_repetition
       detail: '\(q\)-ary repetition code reduce to repetition codes for \(q=2\).'
   cousins:
+    - code_id: mds
+      detail: 'Binary repetition codes are trivial MDS codes \cite[Thm. 12.3.1]{preset:HKSbounds}\cite[Sec. 3.3.2]{preset:HKSclass}.'
     - code_id: perfect_binary
       detail: 'Repetition codes are trivially perfect for odd \(n\) \cite[Def. 12.3.4]{preset:HKSbounds}\cite[pg. 180]{preset:MacSlo}.'
     - code_id: quantum_repetition
diff --git a/codes/classical/bits/reed_muller/dual_hamming/simplex734.yml b/codes/classical/bits/reed_muller/dual_hamming/simplex734.yml
@@ -24,7 +24,7 @@ description: |
   \left(\begin{array}{ccccccc}
    1 & 0 & 1 & 1 & 1 & 0 & 0 \\
    1 & 1 & 1 & 0 & 0 & 1 & 0 \\
-   0 & 1 & 1 & 1 & 0 & 0 & 1 \\
+   0 & 1 & 1 & 1 & 0 & 0 & 1 
   \end{array}\right)~.
   \end{align}
   The automorphism group of the code is \(GL(3,\mathbb{F}_2)\cong PSL(2,\mathbb{F}_7)\), the second-smallest non-abelian finite simple group.
diff --git a/codes/classical/bits/reed_muller/hamming/hamming844.yml b/codes/classical/bits/reed_muller/hamming/hamming844.yml
@@ -24,7 +24,7 @@ description: |
   1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
   0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 \\
   0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 \\
-  0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 \\
+  0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 
   \end{pmatrix}~,
   \end{align}
   equivalent to the standard form
@@ -33,7 +33,7 @@ description: |
   1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 \\
   0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\
   0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 \\
-  0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 \\
+  0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 
   \end{pmatrix}~.
   \end{align}
   Its automorphism group is \(GA(3,\mathbb{F}_2)\) \cite{arxiv:1912.09167}.
@@ -45,9 +45,9 @@ relations:
     - code_id: extended_hamming
     - code_id: biorthogonal
       detail: 'The \([8,4,4]\) extended Hamming code is the first-order RM\((1,3)\) code.'
+  cousins:
     - code_id: group
       detail: 'The \([8,4,4]\) extended Hamming code is a group-algebra code for the group \(\mathbb{Z}_2 \times \mathbb{Z}_4\) \cite{arxiv:1912.09167}.'
-  cousins:
     - code_id: binary_quad_residue
       detail: 'The \([8,4,4]\) extended Hamming code is an extended quadratic-residue code \cite{preset:MacSlo}.'
     - code_id: divisible
diff --git a/codes/classical/bits/reed_muller/reed_muller.yml b/codes/classical/bits/reed_muller/reed_muller.yml
@@ -62,9 +62,9 @@ relations:
       detail: 'RM codes are special cases of hyperbolic evaluation codes \cite[Thm. 3 proof]{doi:10.1007/3-540-45624-4_17}.'
     - code_id: divisible
       detail: 'An RM\((r,m)\) code is \(2^{\left\lceil m/r\right\rceil-1}\)-divisible, according to McEliece''s theorem \cite{doi:10.1016/0097-3165(71)90066-5,doi:10.1016/0012-365X(72)90032-5}.'
+  cousins:
     - code_id: group
       detail: 'RM codes are group-algebra codes \cite{doi:10.1007/BF01072842,preset:Charpin82}\cite[Exam. 16.4.11]{preset:HKSalgebra}. Consider a binary vector space of dimension \( m \). Under addition, this forms a finite group with \( 2^m \) elements known as an elementary Abelian 2-group -- the direct product of \( m \) two-element cyclic groups \( \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2 \). Denote this group by \( G_m \). Let \( J \) be the Jacobson radical of the \hyperref[topic:group-algebra]{group algebra} \( \mathbb{F}_2 G_m \). RM\((r,m)\) codes correspond to the ideal \( J^{m-r} \). The length of the code is \( |G_m| = 2^m \), the distance is \( 2^{m-r} \), and the dimension is \( \sum_{i=0}^r {m \choose i} \). A similar construction exists for choices of a prime \( p\neq 2 \).'
-  cousins:
     - code_id: bch
       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
diff --git a/codes/classical/q-ary_digits/ag/residueAG/shimura.yml b/codes/classical/q-ary_digits/ag/residueAG/shimura.yml
@@ -22,6 +22,7 @@ relations:
   parents:
     - code_id: residue
       detail: 'TVZ codes can be formulated as residue AG codes on algebraic curves.'
+  cousins:
     - code_id: evaluation
       detail: 'TVZ codes can also be formulated as evaluation AG codes on algebraic curves; these are dual to the corresponding residue AG codes.'
 
diff --git a/codes/classical/q-ary_digits/easy/hexacode.yml b/codes/classical/q-ary_digits/easy/hexacode.yml
@@ -50,14 +50,14 @@ relations:
       detail: 'The hexacode is an evaluation AG code over the \hyperref[topic:finite-fields]{quaternary Galois field} \(\mathbb{F}_4 = \{0,1,\omega, \bar{\omega}\}\) with \(\cal X\) defined by \(x^2 y + \omega y^2 z + \bar{\omega} z^2 x = 0\) \cite[Exam. 2.77]{preset:HPAlgCodes}.'
     - code_id: denniston
       detail: 'A version of the hexacode is recovered for Denniston code parameters \(i=1\) and \(a=2\) \cite{doi:10.1201/9781315371993}.'
-    - code_id: mds
-      detail: 'The hexacode is an MDS code \cite[Exer. 578]{doi:10.1017/CBO9780511807077}.'
     - code_id: extended_reed_solomon
       detail: 'The hexacode is a triply extended RS code \cite[pg. 82]{doi:10.1007/978-1-4757-6568-7}.'
     - code_id: lexicographic
       detail: 'Hexacodewords can be arranged in an order from smallest to largest, with each codeword differing at four places from the next \cite{manual:{R. A. Wilson, On lexicographic codes of minimal distance 4, Atti Sem. Mat. Fis. Univ. Modena 33 (1984)}}\cite[pg. 327]{preset:MacSlo}.'
     - code_id: small_distance
   cousins:
+    - code_id: mds
+      detail: 'The hexacode is an MDS code \cite[Exer. 578]{doi:10.1017/CBO9780511807077}.'
     - code_id: q-ary_quad_residue
       detail: 'The hexacode is the smallest example of an extended quadratic-residue code of Type \(4^H\) \cite[Sec. 2.4.6]{doi:10.1007/3-540-30731-1}\cite[Exer. 363]{doi:10.1017/CBO9780511807077}.'
     - code_id: q-ary_hamming
diff --git a/codes/classical/q-ary_digits/easy/ml/one_vs_one.yml b/codes/classical/q-ary_digits/easy/ml/one_vs_one.yml
@@ -23,7 +23,7 @@ description: |
   1 & 1 & 1 & 0 & 0 & 0 \\
   -1 & 0 & 0 & 1 & 1 & 0 \\
   0 & -1 & 0 & -1 & 0 & 1 \\
-  0 & 0 & -1 & 0 & -1 & -1 \\
+  0 & 0 & -1 & 0 & -1 & -1 
   \end{array}
   \right)~.
   \end{align}
diff --git a/codes/classical/q-ary_digits/easy/q-ary_parity_check.yml b/codes/classical/q-ary_digits/easy/q-ary_parity_check.yml
@@ -22,11 +22,11 @@ relations:
       detail: 'RS codes for \(k=n-1\) are parity-check codes \cite{preset:Schurer15}.'
     - code_id: q-ary_cyclic
       detail: 'Since permutations preserve coordinate sums, the cyclic permutation of an SPC codeword is another codeword. The generator polynomial of the code is \(x-1\).'
-    - code_id: mds
     - code_id: checksum
       detail: 'The \(q\)-ary parity check code is a simple example of a checksum code, with the parity of the message being the checksum.'
     - code_id: small_distance
   cousins:
+    - code_id: mds
     - code_id: q-ary_ldgm
       detail: 'Concatenated \(q\)-ary parity-check codes are LDGM \cite{doi:10.1109/20.917609}.'
 
diff --git a/codes/classical/q-ary_digits/easy/q-ary_repetition.yml b/codes/classical/q-ary_digits/easy/q-ary_repetition.yml
@@ -29,6 +29,7 @@ relations:
       detail: 'The \(q\)-ary repetition code is cyclic with generator polynomial \(1+x+\cdots+x^{n-1}\).'
     - code_id: delsarte_optimal_q-ary
       detail: 'The \(q\)-ary repetition code is a \(q\)-ary sharp configuration \cite[Table 12.1]{preset:HKSbounds}.'
+  cousins:
     - code_id: locally_recoverable
       detail: 'The \(q\)-ary repetition code is an LRC with \(r=2\) \cite{arxiv:2311.08653}.'
 
diff --git a/codes/classical/q-ary_digits/easy/ternary_golay.yml b/codes/classical/q-ary_digits/easy/ternary_golay.yml
@@ -24,7 +24,7 @@ description: |
     0 & 0 & 1 & 0 & 0 & 0 & 1 & 2 & 1 & 0 & 2 \\
     0 & 0 & 0 & 1 & 0 & 0 & 2 & 1 & 0 & 1 & 2 \\
     0 & 0 & 0 & 0 & 1 & 0 & 2 & 0 & 1 & 2 & 1 \\
-    0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 2 & 1 & 1 \\
+    0 & 0 & 0 & 0 & 0 & 1 & 0 & 2 & 2 & 1 & 1 
   \end{array}\right)~.
   \end{align}
 
diff --git a/codes/classical/q-ary_digits/easy/tetracode.yml b/codes/classical/q-ary_digits/easy/tetracode.yml
@@ -33,13 +33,13 @@ relations:
       detail: 'The tetracode is equivalent to \(S(3,2)\).'
     - code_id: q-ary_hamming
       detail: 'The tetracode is equivalent to the \(r=2\) \(3\)-ary Hamming code.'
-    - code_id: mds
-      detail: 'The tetracode is a unique MDS code \cite{manual:{O. Taussky and J. Todd, "Covering theorems for groups." Bulletin of the American Mathematical Society. Vol. 54. No. 3. 201 CHARLES ST, PROVIDENCE, RI 02940-2213: AMER MATHEMATICAL SOC, 1948},doi:10.1112/jlms/s1-44.1.60}.'
     - code_id: extended_reed_solomon
       detail: 'The tetracode is an extended RS code \cite[pg. 81]{doi:10.1007/978-1-4757-6568-7}.'
     - code_id: lexicographic
       detail: 'The tetracode is a lexicode \cite{doi:10.1109/TIT.1986.1057187}.'
   cousins:
+    - code_id: mds
+      detail: 'The tetracode is a unique MDS code \cite{manual:{O. Taussky and J. Todd, "Covering theorems for groups." Bulletin of the American Mathematical Society. Vol. 54. No. 3. 201 CHARLES ST, PROVIDENCE, RI 02940-2213: AMER MATHEMATICAL SOC, 1948},doi:10.1112/jlms/s1-44.1.60}.'
     - code_id: ternary_golay
       detail: 'Extended ternary Golay codewords can be obtained from tetracodewords \cite{doi:10.1007/978-1-4757-6568-7}.
       The tetracode can be used to decode the extended ternary Golay code \cite{doi:10.1109/TIT.1986.1057197}.'
diff --git a/codes/classical/rings/over_zq/over_z4/linear_over_z4/self_dual/octacode.yml b/codes/classical/rings/over_zq/over_z4/linear_over_z4/self_dual/octacode.yml
@@ -21,7 +21,7 @@ description: |
     1 & 1 & 2 & 1 & 3 & 0 & 0 & 0 \\
     1 & 0 & 1 & 2 & 1 & 3 & 0 & 0 \\
     1 & 0 & 0 & 1 & 2 & 1 & 3 & 0 \\
-    1 & 0 & 0 & 0 & 1 & 2 & 1 & 3 \\
+    1 & 0 & 0 & 0 & 1 & 2 & 1 & 3 
     \end{array}
     \right)\,,
   \end{align}
@@ -31,7 +31,7 @@ description: |
     \begin{array}{ccccccc}
     1 & 1 & 3 & 2 & 1 & 0 & 0 \\
     0 & 1 & 1 & 3 & 2 & 1 & 0 \\
-    0 & 0 & 1 & 1 & 3 & 2 & 1 \\
+    0 & 0 & 1 & 1 & 3 & 2 & 1 
     \end{array}
     \right)\,.
   \end{align}
diff --git a/codes/classical/spherical/polytope/2d/polygon.yml b/codes/classical/spherical/polytope/2d/polygon.yml
@@ -24,9 +24,9 @@ relations:
     - code_id: self_dual_polytope
       detail: 'Polygons are self-dual.'
     - code_id: sharp_config
+  cousins:
     - code_id: spherical_design
       detail: 'A \(q\)-gon is a tight spherical \(q-1\) design.'
-  cousins:
     - code_id: cat
       detail: 'The \(q(S+1)\)-component cat coherent-state constellation forms the vertices of a \(q(S+1)\)-gon.'
     - code_id: quantum_psk
diff --git a/codes/classical/spherical/polytope/3d/icosahedron.yml b/codes/classical/spherical/polytope/3d/icosahedron.yml
@@ -27,9 +27,9 @@ relations:
     - code_id: polyhedron
     - code_id: sharp_config
       detail: 'The icosahedron is a sharp configuration \cite{manual:{N. N. Andreev, "An extremal property of the icosahedron." East J. Approx 2.4 (1996): 459-462},arxiv:math/0607446}.'
+  cousins:
     - code_id: spherical_design
       detail: 'The icosahedron code forms a unique tight spherical 5-design \cite{doi:10.1007/BF03187604}\cite[Exam. 9.6.1]{preset:EricZin}.'
-  cousins:
     - code_id: dual_polytope
       detail: 'The icosahedron and dodecahedron are dual to each other.'
 
diff --git a/codes/classical/spherical/polytope/4d/24cell/24cell.yml b/codes/classical/spherical/polytope/4d/24cell/24cell.yml
@@ -43,9 +43,9 @@ relations:
       detail: 'The 24-cell code is the minimal shell of the \(D_4\) lattice.'
     - code_id: sidelnikov
       detail: 'The 24-cell code is equivalent to the real Clifford subgroup-orbit code for \(n=4\).'
+  cousins:
     - code_id: spherical_design
       detail: 'The 24-cell code is a spherical 5-design \cite{arxiv:math/0607447}.'
-  cousins:
     - code_id: group_classical
       detail: 'The 24-cell code has a quaternion-coordinate realization as the 24 elements of the binary tetrahedral group \(2T\), one of the three exceptional finite subgroups of \(SU(2)\) \cite{arxiv:2205.04965}.'
     - code_id: 600cell
diff --git a/codes/classical/spherical/polytope/6d/hessian_polyhedron/hessian_polyhedron.yml b/codes/classical/spherical/polytope/6d/hessian_polyhedron/hessian_polyhedron.yml
@@ -44,9 +44,9 @@ relations:
       detail: 'The \(2_{21}\) polytope is self-dual \cite{doi:10.2307/2371466}.'
     - code_id: cgs_spherical
       detail: 'The CGS isotropic subspace code for \(q=2\) reduces to the Hessian polytope.'
+  cousins:
     - code_id: spherical_design
       detail: 'The Hessian polytope code forms a tight spherical 4-design \cite[Exam. 7.3]{arxiv:1104.4692}. The double Hessian polytope code forms a spherical 5-design \cite{preset:Venkov01}.'
-  cousins:
     - code_id: esix
       detail: 'The 27 Hessian polyhedron codewords are intimately related to the \(E_6\) Lie group \cite{arxiv:math/0507118}.'
     - code_id: esix_shell
diff --git a/codes/classical/spherical/polytope/8d/witting_polytope/witting_polytope.yml b/codes/classical/spherical/polytope/8d/witting_polytope/witting_polytope.yml
@@ -48,11 +48,11 @@ relations:
     #  Also in eeight_shell
     - code_id: sharp_config
       detail: 'The Witting polytope code is a sharp configuration \cite{doi:10.1515/dma.1994.4.2.143,arxiv:math/0607446}.'
-    - code_id: spherical_design
-      detail: 'The Witting polytope code forms a tight spherical 7-design \cite{doi:10.4153/CJM-1981-038-7}\cite[Ch. 14]{doi:10.1007/978-1-4757-6568-7}.'
     - code_id: sidelnikov
       detail: 'The Witting polytope code is equivalent to the real Clifford subgroup-orbit code for \(n=8\).'
   cousins:
+    - code_id: spherical_design
+      detail: 'The Witting polytope code forms a tight spherical 7-design \cite{doi:10.4153/CJM-1981-038-7}\cite[Ch. 14]{doi:10.1007/978-1-4757-6568-7}.'
     - code_id: complex_projective
       detail: 'Antipodal pairs of points of the Witting polytope code correspond to the 120 tritangent planes of a canonical sextic curve in \(\mathbb{C}P^3\) \cite{doi:10.1112/plms/s2-35.1.23,preset:Arnold99,arxiv:math/0607446,arxiv:1505.06742}.'
     - code_id: delsarte_optimal
diff --git a/codes/classical/spherical/polytope/infinite/biorthogonal_spherical.yml b/codes/classical/spherical/polytope/infinite/biorthogonal_spherical.yml
@@ -38,12 +38,12 @@ relations:
     - code_id: polytope
       detail: 'Biorthogonal spherical codewords in 2 (3, 4, \(n\)) dimensions form the vertices of a square (octahedron, 16-cell, \(n\)-orthoplex).'
     - code_id: sharp_config
-    - code_id: spherical_design
-      detail: 'Biorthogonal spherical codes are the only tight spherical 3-designs \cite[Tab. 9.3]{preset:EricZin}. A suitable weighted union of the vertices of a hypercube and an orthoplex forms a weighted spherical 5-design in dimensions \(\geq 3\) \cite[Sec. 8.6, Ex. 5-2]{preset:Stroud71}\cite[Exam. 2.6]{arxiv:2403.07457}.'
     - code_id: lattice_shell
       detail: 'Biorthogonal codewords form the minimal shell of the \(\mathbb{Z}^n\) hypercubic lattice.'
     - code_id: permutation_spherical
   cousins:
+    - code_id: spherical_design
+      detail: 'Biorthogonal spherical codes are the only tight spherical 3-designs \cite[Tab. 9.3]{preset:EricZin}. A suitable weighted union of the vertices of a hypercube and an orthoplex forms a weighted spherical 5-design in dimensions \(\geq 3\) \cite[Sec. 8.6, Ex. 5-2]{preset:Stroud71}\cite[Exam. 2.6]{arxiv:2403.07457}.'
     - code_id: 24cell
       detail: 'Vertices of a 24-cell can be split up into vertices of three 16-cells, which are biorthogonal spherical codes for \(n=4\) \cite{preset:coxeter}. The vertices of a 24-cell are a union of the vertices of a tesseract and a 16-cell \cite[Exam. 2.6]{arxiv:2403.07457}.'
     - code_id: dual_polytope
diff --git a/codes/classical/spherical/polytope/infinite/simplex_spherical/simplex_spherical.yml b/codes/classical/spherical/polytope/infinite/simplex_spherical/simplex_spherical.yml
@@ -36,11 +36,11 @@ relations:
     - code_id: self_dual_polytope
       detail: 'The simplex is self-dual.'
     - code_id: sharp_config
-    - code_id: spherical_design
-      detail: 'Simplex spherical codes are the only tight spherical 2-designs \cite[Tab. 9.3]{preset:EricZin}. The bi-simplex is a spherical 3-design since antipodal codes have zero averages over odd-degree polynomials.'
     # QSC paper has design strength 2 for bi-simplex, but it needs to be 3
     - code_id: permutation_spherical
   cousins:
+    - code_id: spherical_design
+      detail: 'Simplex spherical codes are the only tight spherical 2-designs \cite[Tab. 9.3]{preset:EricZin}. The bi-simplex is a spherical 3-design since antipodal codes have zero averages over odd-degree polynomials.'
     - code_id: dodecahedron
       detail: 'Vertices of a dodecahedron can be split up into vertices of five tetrahedra, which are simplex spherical codes for \(n=3\) \cite{preset:coxeter}.'
     - code_id: binary_antipodal
diff --git a/codes/quantum/groups/rotors/stabilizer/css/rotor_3_1_2.yml b/codes/quantum/groups/rotors/stabilizer/css/rotor_3_1_2.yml
@@ -28,10 +28,10 @@ relations:
       detail: 'Taking \(H_X=\begin{pmatrix}-3 & 1 & 2\end{pmatrix}\) and \(H_Z=\begin{pmatrix}4&6&3\end{pmatrix}\) yields the three-rotor code.'
     - code_id: ame
       detail: 'Three-rotor codewords are CV AME states \cite{arxiv:2503.15698}.'
-    - code_id: covariant
-      detail: 'The three-rotor code is \(U(1)\)-covariant.'
     - code_id: small_distance_quantum
   cousins:
+    - code_id: covariant
+      detail: 'The three-rotor code is \(U(1)\)-covariant.'
     - code_id: stab_3_1_2
       detail: 'The three-rotor code is a rotor analogue of the three-qutrit code.'
 
diff --git a/codes/quantum/groups/rotors/stabilizer/rotor_5_1_3.yml b/codes/quantum/groups/rotors/stabilizer/rotor_5_1_3.yml
@@ -20,13 +20,13 @@ protection: |
 relations:
   parents:
     - code_id: rotor_stabilizer
-    - code_id: covariant
-      detail: 'The five-rotor code is \(U(1)\)-covariant.'
     - code_id: ame
       detail: 'Five-rotor codewords are CV AME \cite{arxiv:2503.15698}.'
     - code_id: quantum_cyclic
     - code_id: small_distance_quantum
   cousins:
+    - code_id: covariant
+      detail: 'The five-rotor code is \(U(1)\)-covariant.'
     - code_id: qudit_5_1_3
       detail: 'The five-rotor code is a rotor analogue of the five-qudit code.'
 
diff --git a/codes/quantum/properties/block/tensor_network/quantum_lego.yml b/codes/quantum/properties/block/tensor_network/quantum_lego.yml
diff --git a/codes/quantum/qubits/nonstabilizer/clifford_hierarchy/xp/double_semion_string_net.yml b/codes/quantum/qubits/nonstabilizer/clifford_hierarchy/xp/double_semion_string_net.yml
diff --git a/codes/quantum/qubits/small_distance/small/4/css_4_1_2/css_4_1_2.svg b/codes/quantum/qubits/small_distance/small/4/css_4_1_2/css_4_1_2.svg
diff --git a/codes/quantum/qubits/small_distance/small/4/stab_4_1_2/stab_4_1_2.svg b/codes/quantum/qubits/small_distance/small/4/stab_4_1_2/stab_4_1_2.svg
diff --git a/codes/quantum/qubits/small_distance/small/5/stab_5_1_2/stab_5_1_2.yml b/codes/quantum/qubits/small_distance/small/5/stab_5_1_2/stab_5_1_2.yml
diff --git a/codes/quantum/qubits/small_distance/small/5/stab_5_1_3.yml b/codes/quantum/qubits/small_distance/small/5/stab_5_1_3.yml
diff --git a/codes/quantum/qubits/small_distance/small/7/steane/steane.yml b/codes/quantum/qubits/small_distance/small/7/steane/steane.yml
diff --git a/codes/quantum/qubits/stabilizer/magic/k-orthogonal/quantum_k-orthogonal.yml b/codes/quantum/qubits/stabilizer/magic/k-orthogonal/quantum_k-orthogonal.yml
diff --git a/codes/quantum/qudits_galois/small/css_5_1_3.yml b/codes/quantum/qudits_galois/small/css_5_1_3.yml
