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testability + 3d surface
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valbert4 committed Nov 9, 2023
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2 changes: 1 addition & 1 deletion codes/classical/properties/ltc/ltc.yml
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A technical definition for codes over binary alphabets is provided as follows; for general alphabets, see Ref. \cite{doi:10.1007/978-3-642-16367-8_6}.
The idea behind LTCs is to be able to reliably test whether a given bit-string \(x\) is in the code by only sampling subsets of \(u\) bits.
To have something to check against, we first have to define a collection of length-\(u\) subsets \(S\) of bit locations that are called \textit{allowed local views}.
A code is LTC if the following two conditions are satisfied \cite[Thm. 1.1]{arxiv:2111.04808}.
A code is an LTC if the following two conditions are satisfied \cite[Thm. 1.1]{arxiv:2111.04808}.
First, if \(x\) is a codeword, then all of its restrictions \(x|_S\) to the subsets \(S\) are allowed local views,
\begin{align}
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- code_id: q-ary_cyclic
detail: 'GRM codes with nonzero evaluation points are cyclic (\cite{doi:10.1007/978-94-011-3810-9}, pg. 52).'
- code_id: q-ary_ltc
detail: 'GRM codes for \(r<q\) can be LTCs in the low- \cite{doi:10.1145/103418.103428,doi:10.1145/278298.278306} and high-error \cite{doi:10.1145/258533.258641,doi:10.1145/258533.258642} regimes.'
detail: 'GRM codes for \(r<q\) can be LTCs in the low- \cite{doi:10.1145/103418.103428,doi:10.1145/278298.278306} and high-error \cite{doi:10.1145/258533.258641,doi:10.1145/258533.258642} regimes. They admit weakly stable presentations of their correspoding groups \cite{arxiv:2311.04681}.'
- code_id: group
detail: 'GRM codes over prime-power fields are group-algebra codes \cite{doi:10.1007/BF01072842,manual:{Charpin, Pascale. Codes idéaux de certaines algèbres modulaires. Diss. 1982.},doi:10.1007/BF00141972}\cite[Ex. 16.4.11]{preset:HKSalgebra}.'

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4 changes: 4 additions & 0 deletions codes/classical/q-ary_digits/ltc/q-ary_ltc.yml
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\frac{1}{r}|H x| \geq \frac{R}{n} D(x,C)
\end{align}
holds for any \(q\)-ary string \(x\), where \(D(x,C)\) is the \(q\)-ary Hamming distance between \(x\) and the closest codeword to \(x\) \cite[Def. 11]{arxiv:1911.03069}.
A code satisfying the above constraint without the weight-\(u\) restriction is called an \(R\)\textit{-testable code} \cite{arxiv:2311.04681}.
notes:
- 'Testable binary codes admit weakly stable presentations of their correspoding groups; this property is relevant to the disproof of the Connes embedding conjecture \cite{arxiv:2311.04681}.'

relations:
parents:
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name: '3D surface code'
introduced: '\cite{arxiv:quant-ph/0110143,arxiv:cond-mat/0411752}'

alternative_names:
- '3D toric code'

description: |
A variant of the Kitaev surface code on a 3D lattice. The closely related \textit{solid code} \cite{arxiv:1406.4227} consists of several 3D surface codes stitched together in a way that the distance scales faster than the linear size of the system.
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parents:
- code_id: higher_dimensional_surface
- code_id: quantum_double_abelian
cousins:
- code_id: single_shot
detail: 'Some 3D surface codes are single-shot codes \cite{arxiv:2307.08118}.'


# Begin Entry Meta Information
_meta:
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name: '3D subsystem surface code'
introduced: '\cite{arxiv:2106.02621}'

alternative_names:
- '3D subsystem toric code'

description: |
Subsystem generalization of the surface code on a 3D cubic lattice with stabilizer generators of weight at most three.
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parents:
- code_id: subsystem_stabilizer
- code_id: single_shot
detail: 'The 3D subsystem surface code is a single-shot code \cite{arxiv:2106.02621,arxiv:2305.06365}.'
detail: 'The 3D subsystem surface code is a single-shot code \cite{arxiv:2106.02621,arxiv:2305.06365}; see Ref. \cite{arxiv:2307.08118} for an alternative formulation.'
cousins:
- code_id: 3d_surface
- code_id: self_correct
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