From 1c4edfe0f42d7dc79f7118e7b52180437a70dba3 Mon Sep 17 00:00:00 2001 From: "Victor V. Albert" Date: Fri, 15 Mar 2024 18:48:19 -0400 Subject: [PATCH] [48,24,12] --- codes/classical/bits/cyclic/golay.yml | 4 +-- .../bits/cyclic/self_dual_48_24_12.yml | 34 +++++++++++++++++++ .../block/symmetry/twisted/constacyclic.yml | 1 + .../classical/q-ary_digits/dual/self_dual.yml | 7 +++- codes/quantum/groups/group_quantum.yml | 11 +++++- .../quantum/properties/block/quantum_mds.yml | 6 ++-- .../stabilizer/css/galois_css.yml | 2 ++ 7 files changed, 58 insertions(+), 7 deletions(-) create mode 100644 codes/classical/bits/cyclic/self_dual_48_24_12.yml diff --git a/codes/classical/bits/cyclic/golay.yml b/codes/classical/bits/cyclic/golay.yml index 163d391d9..895475399 100644 --- a/codes/classical/bits/cyclic/golay.yml +++ b/codes/classical/bits/cyclic/golay.yml @@ -67,9 +67,9 @@ relations: - code_id: nearly_perfect detail: 'The extended Golay code is nearly perfect.' - code_id: self_dual - detail: 'The extended Golay code is self-dual.' + detail: 'The extended Golay code is the unique code at its parameters and happens to be self-dual \cite[Remark 4.3.11]{preset:HPRainsSloane}.' - code_id: group - detail: 'The extended Golay code is a group-algebra code for various groups \cite{doi:10.1109/TIT.2008.928260,doi:10.1007/s10623-017-0440-7,doi:10.1016/0097-3165(90)90069-9}\cite[Ex. 16.5.1]{preset:HKSalgebra}.' + detail: 'The extended Golay code is a group-algebra code for various groups \cite{doi:10.1109/TIT.2008.928260,doi:10.1007/s10623-017-0440-7,doi:10.1016/0097-3165(90)90069-9}; see \cite[Ex. 16.5.1]{preset:HKSalgebra}.' - code_id: univ_opt_q-ary detail: 'The Golay code and several of its extended, shortened, and punctured versions are LP universally optimal codes \cite{arxiv:1212.1913}.' diff --git a/codes/classical/bits/cyclic/self_dual_48_24_12.yml b/codes/classical/bits/cyclic/self_dual_48_24_12.yml new file mode 100644 index 000000000..5fbfb9fa2 --- /dev/null +++ b/codes/classical/bits/cyclic/self_dual_48_24_12.yml @@ -0,0 +1,34 @@ +####################################################### +## This is a code entry in the error correction zoo. ## +## https://github.com/errorcorrectionzoo ## +####################################################### + +code_id: self_dual_48_24_12 +physical: bits +logical: bits + +name: '\([48,24,12]\) self-dual code' + +description: | + An extended quadratic-residue code that is known to be the only self-dual doubly-even code at its parameters \cite{doi:10.1109/TIT.2002.806146}. + + The code's automorphism group is \(PSL_2(47)\) \cite[Remark 4.3.11]{preset:HPRainsSloane}. + +relations: + parents: + - code_id: binary_quad_residue + - code_id: self_dual + detail: 'The \([48,24,12]\) self-dual code is the only self-dual doubly-even code at its parameters \cite{doi:10.1109/TIT.2002.806146}.' + - code_id: divisible + detail: 'The \([48,24,12]\) self-dual code is the only self-dual doubly-even code at its parameters \cite{doi:10.1109/TIT.2002.806146}.' + cousins: + - code_id: combinatorial_design + detail: 'Fixed-weight codewords of extremal self-dual doubly-even codes whose length divides 24 form a combinatorial 5-design \cite{doi:10.1016/S0021-9800(69)80115-8}.' + + +# Begin Entry Meta Information +_meta: + # Change log - most recent first + changelog: + - user_id: VictorVAlbert + date: '2024-03-15' diff --git a/codes/classical/properties/block/symmetry/twisted/constacyclic.yml b/codes/classical/properties/block/symmetry/twisted/constacyclic.yml index 36a8f60cc..768239d41 100644 --- a/codes/classical/properties/block/symmetry/twisted/constacyclic.yml +++ b/codes/classical/properties/block/symmetry/twisted/constacyclic.yml @@ -13,6 +13,7 @@ alternative_names: description: | A classical code \(C\) of length \(n\) over an alphabet \(R\) is \(\alpha\)-constacyclic (or \(\alpha\)-twisted) if, for each string \(c_1 c_2 \cdots c_n\in C\), the string \(\alpha c_n, c_1, \cdots, c_{n-1} \in C\). + A \(-1\)-constacyclic code is called \textit{negacyclic}. relations: diff --git a/codes/classical/q-ary_digits/dual/self_dual.yml b/codes/classical/q-ary_digits/dual/self_dual.yml index e738eb151..3cdd18df1 100644 --- a/codes/classical/q-ary_digits/dual/self_dual.yml +++ b/codes/classical/q-ary_digits/dual/self_dual.yml @@ -32,6 +32,11 @@ protection: | except for \(n = 22\) modulo four for the second case, where the bound is increased by four \cite{doi:10.1109/18.651000}. A self-dual code saturating the above inequality is called \textit{extremal}. + Fixed-weight codewords of extremal self-dual doubly-even binary codes whose length divides 24 form a combinatorial 5-design \cite{doi:10.1016/S0021-9800(69)80115-8}. + The extended Golay code and the \([48,24,12]\) self-dual code are two such codes. + It is not yet known whether a \([72,36,16]\) self-dual code exists; see \cite{doi:10.1109/TIT.1973.1054975,doi:10.1090/conm/634/12692}\cite[Remark 4.3.11]{preset:HPRainsSloane}. + + For ternary self-dual codes, see \cite[Remark 4.3.14]{preset:HPRainsSloane}\cite{doi:10.1016/j.ffa.2005.05.012}. notes: - See books \cite{doi:10.1007/3-540-30731-1,doi:10.1017/CBO9780511807077} for more on self-dual codes. @@ -46,7 +51,7 @@ relations: In addition, quaternary linear codes are Hermitian self-orthogonal (self-dual) iff they are trace-Hermitian self-orthogonal (self-dual) additive \cite[Thm. 27.4.1]{preset:HKSquantum} (\cite[Thm. 9.10.3]{doi:10.1017/CBO9780511807077}).' cousins: - code_id: divisible - detail: 'Binary self-dual codes are singly-even and binary self-orthogonal codes that are not doubly-even are singly-even \cite[Def. 4.1.6]{preset:HKSselfdual}. + detail: 'Binary self-dual codes are singly-even, and binary self-orthogonal codes that are not doubly-even are singly-even \cite[Def. 4.1.6]{preset:HKSselfdual}. The minimum distance of doubly-even binary self-dual codes asymptotically satisfies \(d\leq0.1664n+o(n)\) \cite{doi:10.1109/18.605587}.' diff --git a/codes/quantum/groups/group_quantum.yml b/codes/quantum/groups/group_quantum.yml index 0e7ebc9a0..1a98e0c49 100644 --- a/codes/quantum/groups/group_quantum.yml +++ b/codes/quantum/groups/group_quantum.yml @@ -9,7 +9,16 @@ logical: groups name: 'Group-based quantum code' -description: 'Encodes a \textit{logical} Hilbert space, finite- or infinite-dimensional, into a \textit{physical} Hilbert space of \(\ell^2\)-normalizable functions on a second-countable unimodular group. For \(K\)-dimensional logical subspace and for groups \(G^{n}\), can be denoted as \(((n,K))_G\). When the logical subspace is the Hilbert space of \(\ell^2\)-normalizable functions on \(G^{ k}\), can be denoted as \([[n,k]]_G\). Ideal codewords may not be normalizable, depending on whether \(G\) is continuous and/or noncompact, so approximate versions have to be constructed in practice.' +description: 'Encodes a \textit{logical} Hilbert space, finite- or infinite-dimensional, into a \textit{physical} Hilbert space of \(\ell^2\)-normalizable functions on a second-countable unimodular group \(G\), i.e., a \(G\)\textit{-valued qudit}. +For \(K\)-dimensional logical subspace and for block codes defined on groups \(G^{n}\), can be denoted as \(((n,K))_G\). +When the logical subspace is the Hilbert space of \(\ell^2\)-normalizable functions on \(G^{ k}\), can be denoted as \([[n,k]]_G\). +Ideal codewords may not be normalizable, depending on whether \(G\) is continuous and/or noncompact, so approximate versions have to be constructed in practice.' + + +features: + general_gates: + - 'Various gates for a single \(G\)-valued qudit are described in \cite{arXiv:1911.00099,arxiv:2208.12309}.' + relations: parents: diff --git a/codes/quantum/properties/block/quantum_mds.yml b/codes/quantum/properties/block/quantum_mds.yml index 59cddd2b2..79168c05e 100644 --- a/codes/quantum/properties/block/quantum_mds.yml +++ b/codes/quantum/properties/block/quantum_mds.yml @@ -44,11 +44,11 @@ relations: - code_id: stabilizer_over_gfqsq detail: 'Many MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction \cite{arxiv:quant-ph/0312164,arxiv:0906.2509,arxiv:1507.08355,arxiv:1803.07927}, in particular from cyclic \cite{doi:10.1109/TIT.2011.2159039}, constacyclic \cite{doi:10.1109/TIT.2014.2308180,doi:10.1109/TIT.2015.2388576,arxiv:1803.07927} and negacyclic \cite{doi:10.1109/TIT.2012.2220519} codes.' - code_id: constacyclic - detail: 'Many MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction from constacyclic codes \cite{doi:10.1109/TIT.2014.2308180,doi:10.1109/TIT.2015.2388576,arxiv:1803.07927}.' + detail: 'Many MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction \cite{arxiv:quant-ph/0312164,arxiv:0906.2509,arxiv:1507.08355,arxiv:1803.07927}, in particular from cyclic \cite{doi:10.1109/TIT.2011.2159039}, constacyclic \cite{doi:10.1109/TIT.2014.2308180,doi:10.1109/TIT.2015.2388576,arxiv:1803.07927}, and negacyclic \cite{doi:10.1109/TIT.2012.2220519} codes.' - code_id: generalized_reed_solomon detail: 'Some MDS codes are constructed from cyclic and constacyclic codes \cite{arxiv:1502.05267} which are GRS codes \cite{doi:10.1007/s10623-022-01174-5,doi:10.1007/s10623-023-01294-6}.' - - code_id: cyclic - detail: 'Many MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction \cite{arxiv:quant-ph/0312164,arxiv:0906.2509,arxiv:1507.08355,arxiv:1803.07927}, in particular from cyclic codes \cite{doi:10.1109/TIT.2011.2159039}.' +# - code_id: cyclic +# detail: 'Many MDS codes are constructed from Hermitian self-orthogonal codes over \(GF(q^2)\) using the Hermitian construction \cite{arxiv:quant-ph/0312164,arxiv:0906.2509,arxiv:1507.08355,arxiv:1803.07927}, in particular from cyclic codes \cite{doi:10.1109/TIT.2011.2159039}.' # Begin Entry Meta Information diff --git a/codes/quantum/qudits_galois/stabilizer/css/galois_css.yml b/codes/quantum/qudits_galois/stabilizer/css/galois_css.yml index ee2abd053..873f27430 100644 --- a/codes/quantum/qudits_galois/stabilizer/css/galois_css.yml +++ b/codes/quantum/qudits_galois/stabilizer/css/galois_css.yml @@ -42,6 +42,8 @@ description: | |\gamma + C_Z^\perp \rangle = \frac{1}{\sqrt{|C_Z^\perp|}} \sum_{\eta \in C_Z^\perp} |\gamma + \eta\rangle. \end{align} + Galois-qudit CSS codes can also be understood in terms graphs via the \textit{reflexive stabilizer} framework, which also allows one to define a code for a given set of Pauli errors \cite{arxiv:2110.08414}. + protection: 'Detects errors on \(d-1\) qubits, corrects errors on \(\left\lfloor (d-1)/2 \right\rfloor\) qubits.' relations: