From 8d9ac4ad1bbe3729b8efa6cd4798d880e0ae8794 Mon Sep 17 00:00:00 2001 From: VVA2024 Date: Thu, 22 Aug 2024 18:24:26 -0400 Subject: [PATCH] ~ --- codes/classical/bits/cyclic/binary_quad_residue.yml | 2 ++ codes/classical/bits/cyclic/golay.yml | 2 +- codes/classical/bits/easy/hamming/hamming.yml | 2 ++ .../classical/q-ary_digits/ag/rm/generalized_reed_muller.yml | 2 +- codes/quantum/properties/block/topological/topological.yml | 5 +++-- 5 files changed, 9 insertions(+), 4 deletions(-) diff --git a/codes/classical/bits/cyclic/binary_quad_residue.yml b/codes/classical/bits/cyclic/binary_quad_residue.yml index e40a2b04c..bf8c98f77 100644 --- a/codes/classical/bits/cyclic/binary_quad_residue.yml +++ b/codes/classical/bits/cyclic/binary_quad_residue.yml @@ -16,6 +16,8 @@ description: | The roots of the generator polynomial \(r(x)\) of the first code (see \ref{topic:Cyclic-to-polynomial-correspondence}) are all of the inequivalent quadratic residues of \(n\), and the second code's generator polynomial is \((x-1)r(x)\). The roots of the generator polynomial \(a(x)\) of the third code are all inequivalent nonresidues of \(n\), and the fourth code's generator polynomial is \((x-1)a(x)\). The codes corresponding to polynomials \(r,a\) are often called \textit{augmented} quadratic-residue codes, while the remaining codes are called \textit{expurgated}. + Their automorphism group is either \(PSL(2,GF(p))\) or a closely related group by the Gleason-Prange theorem \cite{doi::10.1109/18.133245,preset:MacSlo}. + features: decoders: - 'Algebraic decoder \cite{manual:{Chen, Y. H., Truong, T. K., Chang, Y., Lee, C. D., & Chen, S. H. (2007). Algebraic decoding of quadratic residue codes using Berlekamp-Massey algorithm. Journal of information science and engineering, 23(1), 127-145.}}.' diff --git a/codes/classical/bits/cyclic/golay.yml b/codes/classical/bits/cyclic/golay.yml index 6a948fda6..0b39761c8 100644 --- a/codes/classical/bits/cyclic/golay.yml +++ b/codes/classical/bits/cyclic/golay.yml @@ -12,7 +12,7 @@ introduced: '\cite{manual:{M. J. E. Golay, \emph{Notes on digital coding}, Proc. description: | A \([23, 12, 7]\) perfect binary linear code with connections to various areas of mathematics, e.g., lattices \cite{doi:10.1007/978-1-4757-6568-7} and sporadic simple groups \cite{preset:MacSlo}. - Adding a parity bit to the code results in the \([24, 12, 8]\) \textit{extended Golay code}. + Adding a parity bit to the code results in the self-dual \([24, 12, 8]\) \textit{extended Golay code}. Up to equivalence, both codes are unique for their respective parameters \cite{doi:10.1016/0012-365X(75)90047-3}. Shortening the Golay code yields the \([22,10,8]\), \([22,11,7]\), and \([22,12,6]\) \textit{shortened Golay codes} \cite{doi:10.1109/18.57203}. The dual of the Golay code is its \([23,11,8]\) even-weight subcode \cite{manual:{W. Feit. Some remarks on weight functions of spaces over GF(2), unpublished (1972)},doi:10.1016/0012-365X(74)90085-5}. diff --git a/codes/classical/bits/easy/hamming/hamming.yml b/codes/classical/bits/easy/hamming/hamming.yml index a768beb64..04d9f814f 100644 --- a/codes/classical/bits/easy/hamming/hamming.yml +++ b/codes/classical/bits/easy/hamming/hamming.yml @@ -38,6 +38,8 @@ relations: - code_id: q-ary_hamming - code_id: narrow_sense_q-ary_bch detail: 'Binary Hamming codes are binary primitive narrow-sense BCH codes \cite[Corr. 5.1.5]{doi:10.1017/CBO9780511807077}. Binary Hamming codes can be written in cyclic form \cite[Thm. 12.22]{preset:Hill}.' + - code_id: bch + detail: 'Binary Hamming codes are binary primitive narrow-sense BCH codes \cite[Corr. 5.1.5]{doi:10.1017/CBO9780511807077}. Binary Hamming codes can be written in cyclic form \cite[Thm. 12.22]{preset:Hill}.' - code_id: lexicographic detail: 'Hamming codes are lexicodes \cite{doi:10.1109/TIT.1986.1057187}.' - code_id: univ_opt_q-ary diff --git a/codes/classical/q-ary_digits/ag/rm/generalized_reed_muller.yml b/codes/classical/q-ary_digits/ag/rm/generalized_reed_muller.yml index 1bf43f3b2..d7ed80613 100644 --- a/codes/classical/q-ary_digits/ag/rm/generalized_reed_muller.yml +++ b/codes/classical/q-ary_digits/ag/rm/generalized_reed_muller.yml @@ -16,7 +16,7 @@ description: | Since \(\beta^q=\beta\) for any \(\beta\in GF(q)\), the above definition is not injective. Replacing each factor in each polynomial as \(x^q\to x\), the above set reduces to the set of all degree-\(\leq r\) polynomials in \(m\) variables such that no term has an exponent \(q\) or higher on any variable. - Its automorphism group is the general affine group \(GA(m,GF(q))\). + Its automorphism group is the general affine group \(GA(m,GF(q))\) \cite{doi:10.1016/0012-365X(93)90321-J}. Any nontrivial \(q\)-ary linear code invariant under this group is equivalent to a GRM code \cite{doi:10.1109/TIT.1970.1054554}. protection: 'Code parameters for specific \(m,r\) are given in Ref. \cite{preset:TVNAlgCodes}\cite[pg. 46]{doi:10.1007/978-94-011-3810-9}.' diff --git a/codes/quantum/properties/block/topological/topological.yml b/codes/quantum/properties/block/topological/topological.yml index e9f8906db..b74883ae3 100644 --- a/codes/quantum/properties/block/topological/topological.yml +++ b/codes/quantum/properties/block/topological/topological.yml @@ -123,10 +123,11 @@ relations: parents: - code_id: block_quantum detail: 'Topological codes are block codes because an infinite family of tensor-product Hilbert spaces is required to formally define a phase of matter.' - cousins: - code_id: hamiltonian detail: 'Codespace of a topological code is typically the ground-state or low-energy subspace of a geometrically local Hamiltonian admitting a topological phase. - Logical qubits can also be created via lattice defects or by appropriately scheduling measurements of gauge generators (see Floquet codes).' + Logical qubits can also be created via lattice defects or by appropriately scheduling measurements of gauge generators (see Floquet codes). + Geometrically local frustration-free code Hamiltonians on Euclidean manifolds are stable with respect to small perturbations when they satisfy the local topological quantum order condition (cf. the \hyperref[topic:tqo]{TQO conditions}), meaning that a notion of a phase can be defined \cite{arxiv:1109.1588,arxiv:2110.11194}.' + cousins: - code_id: cluster_state detail: 'There exist necessary and sufficient conditions for a family of cluster states to exhibit the TQO-1 property \cite{arxiv:2112.02502}.'