From 7e7d63b883b3f8b4cdee9ccd61c1a3e98f6e138c Mon Sep 17 00:00:00 2001 From: "Victor V. Albert" Date: Tue, 10 Dec 2024 17:13:15 -0500 Subject: [PATCH] dodecahedron + links to it --- codes/classical/bits/cyclic/golay.yml | 4 +++ .../bits/graph/incidence/petersen.yml | 3 ++ .../ag/reed_solomon/berlekamp.yml | 2 +- .../spherical/polytope/dodecahedron.yml | 33 +++++++++++++++++++ .../spherical/polytope/icosahedron.yml | 2 -- .../polytope/infinite/simplex_spherical.yml | 2 +- .../polytope/pentakis_dodecahedron.yml | 2 ++ .../stabilizer/lattice/multimodegkp.yml | 5 ++- .../small/quantum_dodecahedron.yml | 5 ++- .../2d_surface/stellated_dodecahedron_css.yml | 4 ++- .../non-css/rhombic_dodecahedron_surface.yml | 4 ++- 11 files changed, 58 insertions(+), 8 deletions(-) create mode 100644 codes/classical/spherical/polytope/dodecahedron.yml diff --git a/codes/classical/bits/cyclic/golay.yml b/codes/classical/bits/cyclic/golay.yml index 94a9a7af1..c5ec70666 100644 --- a/codes/classical/bits/cyclic/golay.yml +++ b/codes/classical/bits/cyclic/golay.yml @@ -99,6 +99,10 @@ relations: detail: 'The dual of the Golay code forms a spherical three-design under the \hyperref[topic:antipodal-mapping]{antipodal mapping} \cite[Exam. 9.3]{doi:10.1007/BF03187604}.' - code_id: lexicographic detail: 'The extended Golay code is a lexicode \cite{manual:{M. J. T. Guy, unpublished.},doi:10.1109/TIT.1986.1057187}\cite[pg. 327]{preset:MacSlo}.' + - code_id: icosahedron + detail: 'The parity bits of the extended Golay code can be visualized to lie on the vertices of the icosahedron; see \href{https://blogs.ams.org/visualinsight/2015/12/01/golay-code/}{post} by J. Baez for more details. To construct the Golay code, one can use the great dodecahedron to generate codewords by placing message bits on the faces and calculating the parity bits that live on the 12 vertices of the inner icosahedron.' + - code_id: dodecahedron + detail: 'The parity bits of the extended Golay code can be visualized to lie on the vertices of the icosahedron; see \href{https://blogs.ams.org/visualinsight/2015/12/01/golay-code/}{post} by J. Baez for more details. To construct the Golay code, one can use the great dodecahedron to generate codewords by placing message bits on the faces and calculating the parity bits that live on the 12 vertices of the inner icosahedron.' # The Golay code can be constructed as a cyclic code with the generator polynomial \(x^{11} + x^{10} + x^6 + x^5 + x^4 + x^2 + 1\) over \(GF(2)\). diff --git a/codes/classical/bits/graph/incidence/petersen.yml b/codes/classical/bits/graph/incidence/petersen.yml index d74794c4a..895acb51c 100644 --- a/codes/classical/bits/graph/incidence/petersen.yml +++ b/codes/classical/bits/graph/incidence/petersen.yml @@ -21,6 +21,9 @@ relations: parents: - code_id: homological_classical - code_id: small_distance + cousins: + - code_id: dodecahedron + detail: 'The Petersen graph can be thought of as a dodecahedron with antipodes identified \cite[Appx. A.2.1]{arxiv:1703.07847}.' # Begin Entry Meta Information diff --git a/codes/classical/q-ary_digits/ag/reed_solomon/berlekamp.yml b/codes/classical/q-ary_digits/ag/reed_solomon/berlekamp.yml index 91d3f15b0..26b942774 100644 --- a/codes/classical/q-ary_digits/ag/reed_solomon/berlekamp.yml +++ b/codes/classical/q-ary_digits/ag/reed_solomon/berlekamp.yml @@ -24,7 +24,7 @@ relations: - code_id: alternant detail: 'Berlekamp codes reduce to narrow-sense alternant codes for \(p=2\) \cite[Ch. 10.6]{doi:10.1017/CBO9780511808968}.' - code_id: reed_solomon - detail: 'Berlekamp codes are obtained by first constructing an RS-like parity-check matrix out of a certain \hyperref[topic:finite-fields]{field extension} of \(GF(p)\) and then taking the \hyperref[topic:finite-fields]{subfield} subcode of the corresponding code; see \cite[Ch. 10.6]{doi:10.1017/CBO9780511808968}..' + detail: 'Berlekamp codes are obtained by first constructing an RS-like parity-check matrix out of a certain \hyperref[topic:finite-fields]{field extension} of \(GF(p)\) and then taking the \hyperref[topic:finite-fields]{subfield} subcode of the corresponding code; see \cite[Ch. 10.6]{doi:10.1017/CBO9780511808968}.' # Begin Entry Meta Information diff --git a/codes/classical/spherical/polytope/dodecahedron.yml b/codes/classical/spherical/polytope/dodecahedron.yml new file mode 100644 index 000000000..04e73b48b --- /dev/null +++ b/codes/classical/spherical/polytope/dodecahedron.yml @@ -0,0 +1,33 @@ +####################################################### +## This is a code entry in the error correction zoo. ## +## https://github.com/errorcorrectionzoo ## +####################################################### + +code_id: dodecahedron +physical: spheres +logical: reals + +name: 'Dodecahedron code' + +description: | + Spherical \((3,20,2-2\sqrt{5}/2)\) code whose codewords are the vertices of the dodecahedron (alternatively, the centers of the faces of a icosahedron, the dodecahedron's dual polytope). + + +relations: + parents: + - code_id: polytope + - code_id: spherical_design + detail: 'The dodecahedron code forms a spherical 5-design \cite{arxiv:2302.11593}.' + cousins: + - code_id: dual_polytope + detail: 'The icosahedron and dodecahedron are dual to each other.' + - code_id: icosahedron + detail: 'The icosahedron and dodecahedron are dual to each other.' + + +# Begin Entry Meta Information +_meta: + # Change log - most recent first + changelog: + - user_id: VictorVAlbert + date: '2024-12-10' diff --git a/codes/classical/spherical/polytope/icosahedron.yml b/codes/classical/spherical/polytope/icosahedron.yml index 177feeaba..05eb50736 100644 --- a/codes/classical/spherical/polytope/icosahedron.yml +++ b/codes/classical/spherical/polytope/icosahedron.yml @@ -26,8 +26,6 @@ relations: - 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: golay - detail: 'The parity bits of the extended Golay code can be visualized to lie on the vertices of the icosahedron; see \href{https://blogs.ams.org/visualinsight/2015/12/01/golay-code/}{post} by J. Baez for more details.' - code_id: dual_polytope detail: 'The icosahedron and dodecahedron are dual to each other.' diff --git a/codes/classical/spherical/polytope/infinite/simplex_spherical.yml b/codes/classical/spherical/polytope/infinite/simplex_spherical.yml index de79798c3..efd3ff22a 100644 --- a/codes/classical/spherical/polytope/infinite/simplex_spherical.yml +++ b/codes/classical/spherical/polytope/infinite/simplex_spherical.yml @@ -29,7 +29,7 @@ relations: detail: 'Simplex spherical codes are the only tight spherical 2-designs \cite[Tab. 9.3]{preset:EricZin}.' - code_id: permutation_spherical cousins: - - code_id: icosahedron + - 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 detail: 'Binary simplex codes map to \((2^m,2^m+1)\) simplex spherical codes under the \hyperref[topic:antipodal-mapping]{antipodal mapping} \cite[Sec. 6.5.2]{manual:{Forney, G. D. (2003). 6.451 Principles of Digital Communication II, Spring 2003.}}\cite[pg. 18]{preset:EricZin}. In other words, simplex (simplex spherical) codes form simplices in Hamming (Euclidean) space.' diff --git a/codes/classical/spherical/polytope/pentakis_dodecahedron.yml b/codes/classical/spherical/polytope/pentakis_dodecahedron.yml index 4edfb546a..3bea798d8 100644 --- a/codes/classical/spherical/polytope/pentakis_dodecahedron.yml +++ b/codes/classical/spherical/polytope/pentakis_dodecahedron.yml @@ -21,6 +21,8 @@ relations: cousins: - code_id: icosahedron detail: 'The pentakis dodecahedron is the convex hull of the icosahedron and dodecahedron.' + - code_id: dodecahedron + detail: 'The pentakis dodecahedron is the convex hull of the icosahedron and dodecahedron.' # Begin Entry Meta Information diff --git a/codes/quantum/oscillators/stabilizer/lattice/multimodegkp.yml b/codes/quantum/oscillators/stabilizer/lattice/multimodegkp.yml index 0618a69cc..fb37bd603 100644 --- a/codes/quantum/oscillators/stabilizer/lattice/multimodegkp.yml +++ b/codes/quantum/oscillators/stabilizer/lattice/multimodegkp.yml @@ -22,7 +22,10 @@ description: | protection: 'The level of protection against displacement errors is quantified by the Euclidean code distance \(\Delta=\min_{x\in {\mathcal{L}}^{\perp}\setminus {\mathcal{L}}} \|x\|_2\) \cite{arxiv:2109.14645}.' features: - rate: 'Transmission schemes with multimode GKP codes achieve, up to a constant-factor offset, the capacity of \hyperref[topic:ad]{AD}, a lower bound on displacement-noise, and a lower bound on thermal-noise Gaussian channel capacities \cite{arxiv:quant-ph/0105058,arxiv:1708.07257,arxiv:1801.04731,arxiv:1801.07271}. Particular random lattice families of multimode GKP codes achieve the hashing bound of the displacement noise channel \cite{arxiv:quant-ph/0105058}.' + rate: | + Transmission schemes with multimode GKP codes achieve a lower bound on displacement noise and a lower bound on the thermal-noise Gaussian channel capacities \cite{arxiv:quant-ph/0105058,arxiv:1708.07257,arxiv:1801.04731,arxiv:1801.07271}. + Particular random lattice families of multimode GKP codes achieve the hashing bound of the displacement noise channel \cite{arxiv:quant-ph/0105058}. + Particular families of GKP codes achieve the capacity of \hyperref[topic:ad]{AD} and amplifications channels \cite{arxiv:2412.06715}. encoders: - 'GKP codes with fixed \(n\) and prime-dimensional logical Hilbert space are symplectically related to a disjoint product of single-mode GKP codes on \(n\) modes, such that encoding via Gaussian unitaries is possible.' diff --git a/codes/quantum/qubits/small_distance/small/quantum_dodecahedron.yml b/codes/quantum/qubits/small_distance/small/quantum_dodecahedron.yml index bb4eff379..f71798ae1 100644 --- a/codes/quantum/qubits/small_distance/small/quantum_dodecahedron.yml +++ b/codes/quantum/qubits/small_distance/small/quantum_dodecahedron.yml @@ -11,7 +11,7 @@ name: '\([[16,4,3]]\) dodecahedral code' introduced: '\cite{arxiv:2411.14448}' description: | - A \([[16,4,3]]\) qubit stabilizer code defined whose \hyperref[topic:encoder-respecting]{encoder-respecting form} is the graph of vertices of a dodecahedron \cite{arxiv:2411.14448}.. + A \([[16,4,3]]\) qubit stabilizer code defined whose \hyperref[topic:encoder-respecting]{encoder-respecting form} is the graph of vertices of a dodecahedron \cite{arxiv:2411.14448}. features: rate: 'The code has a rate of \(1/4\), higher than that of the five-qubit perfect code.' @@ -23,6 +23,9 @@ relations: parents: - code_id: qubit_stabilizer - code_id: small_distance_quantum + cousins: + - code_id: dodecahedron + detail: 'The \hyperref[topic:encoder-respecting]{encoder-respecting form} of the \([[16,4,3]]\) dodecahedral code is the graph of vertices of a dodecahedron \cite{arxiv:2411.14448}.' # Begin Entry Meta Information diff --git a/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/stellated_dodecahedron_css.yml b/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/stellated_dodecahedron_css.yml index 7fe6c560b..4915251ef 100644 --- a/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/stellated_dodecahedron_css.yml +++ b/codes/quantum/qubits/stabilizer/topological/surface/2d_surface/stellated_dodecahedron_css.yml @@ -29,7 +29,9 @@ relations: - code_id: polytope detail: 'Bring code and related codes listed in \cite[Table 1]{arxiv:1712.07666} arrange qubits and stabilizer generators on star polyhedra.' - code_id: golay - detail: 'The automorphism group of the parity-check matrix of the Golay code is the same as a certai automorphism group of the Bring code \cite{arxiv:2202.06647}.' + detail: 'The automorphism group of the parity-check matrix of the Golay code is the same as a certain automorphism group of the Bring code \cite{arxiv:2202.06647}.' + - code_id: dodecahedron + detail: 'The qubits and stabilizer generators of the \([[30,8,3]]\) Bring code lie on the vertices of the small stellated dodecahedron.' # Begin Entry Meta Information diff --git a/codes/quantum/qubits/stabilizer/topological/surface/non-css/rhombic_dodecahedron_surface.yml b/codes/quantum/qubits/stabilizer/topological/surface/non-css/rhombic_dodecahedron_surface.yml index ac63f7620..aae441253 100644 --- a/codes/quantum/qubits/stabilizer/topological/surface/non-css/rhombic_dodecahedron_surface.yml +++ b/codes/quantum/qubits/stabilizer/topological/surface/non-css/rhombic_dodecahedron_surface.yml @@ -7,7 +7,7 @@ code_id: rhombic_dodecahedron_surface physical: qubits logical: qubits -name: 'Rhombic dodecahedron surface code' +name: '\([[14,3,3]]\) Rhombic dodecahedron surface code' introduced: '\cite{arxiv:2010.06628}' alternative_names: @@ -25,6 +25,8 @@ relations: cousins: - code_id: polytope detail: 'The rhombic dodecahedron surface code arranges qubits and stabilizer generators on polytopes.' + - code_id: dodecahedron + detail: 'The qubits of the \([[14,3,3]]\) rhombic dodecahedron surface code lie on the vertices of the small stellated dodecahedron.' # Begin Entry Meta Information