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Merge pull request #333 from LeonidPryadko/mod6
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Mod6
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valbert4 authored Oct 18, 2023
2 parents b8e8082 + 7ad63b1 commit 9299a1e
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6 changes: 3 additions & 3 deletions codes/quantum/qudits_galois/qldpc/algebraic/2bga.yml
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description: |
2BGA codes are the smallest \hyperref[code:lifted_product]{LP codes}
LP\((a,b)\), constructed from a pair of \hyperref[topic:group-algebra]{group algebra} elements
LP\((a,b)\), \hyperref[topic:galois_css]{CSS codes} constructed from a pair of \hyperref[topic:group-algebra]{group algebra} elements
\(a,b\in \mathbb{F}_q[G]\), where \(G\) is a finite group, and \(\mathbb{F}_q\) is a Galois field.
For a group of order \(\ell\), we get a 2BGA code of length
\(n=2\ell\).
A 2BGA code for an Abelian group is called an \textit{Abelian 2BGA code}.
An \(\mathbb{F}_q\)-linear code isomorphic to a \(Z\)-part of the 2BGA code LP\((a,b)\) can be most
An \(\mathbb{F}_q\)-linear code isomorphic to a \(Z\) logical subspace of the 2BGA code LP\((a,b)\) can be most
naturally defined as a linear space of pairs
\((u,v)\in \mathbb{F}_q[G]\times \mathbb{F}_q[G]\) such that
\begin{align}
a u+v b=0,
\end{align}
with any two pairs \((u,v)\) and \((u',v')\) such that \(u'=u+w b\)
and \(v'=v-aw\) identified. The order in the products is relevant
and \(v'=v-aw\) considered equivalent. The order in the products is relevant
when the group is non-Abelian.
For example, consider the
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