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enumerate lists in generalized_bicycle
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phfaist committed Oct 16, 2023
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The number of qudits encoded in such a GB code is \(k=2\deg h(x)\), twice the dimension of the underlying classical code \cite{arxiv:1904.02703}.
Two codes GB\((a,b)\) and GB\((a',b')\) of the same size \(n=2\ell\) are equivalent if one of the following conditions is satisfied \cite{arxiv:2203.17216}:
(1) \(a'(x)=a(x^{m})\) mod \(x^{\ell}-1\), \(b'(x)=b(x^{m})\) mod \(x^{\ell}-1\) for some \(m\) mutually prime with \(\ell\), gcd\((m,\ell)=1\);
(2) \(a'(x)=b(x), b'(x)=a(x)\);
(3) \(a'(x)\) and \(b'(x)\) are the reciprocal polynomials of \(a(x)\) and \(b(x)\), respectively;
(4) \(a'(x)=\delta a(x), b'(x)=b(x)\), for some \(0\neq\delta\in GF(q)=\mathbb{F}_q\);
(5) \(a'(x)=f(x)a(x), b'(x)=f(x)b(x)\), for some polynomial \(f(x)\in \mathbb{F}_q[x]\) such that gcd\((f,x^{\ell}-1)=1\).
\begin{enumerate}[(1)]
\item \(a'(x)=a(x^{m})\) mod \(x^{\ell}-1\), \(b'(x)=b(x^{m})\) mod \(x^{\ell}-1\) for some \(m\) mutually prime with \(\ell\), gcd\((m,\ell)=1\);
\item \(a'(x)=b(x), b'(x)=a(x)\);
\item \(a'(x)\) and \(b'(x)\) are the reciprocal polynomials of \(a(x)\) and \(b(x)\), respectively;
\item \(a'(x)=\delta a(x), b'(x)=b(x)\), for some \(0\neq\delta\in GF(q)=\mathbb{F}_q\);
\item \(a'(x)=f(x)a(x), b'(x)=f(x)b(x)\), for some polynomial \(f(x)\in \mathbb{F}_q[x]\) such that gcd\((f,x^{\ell}-1)=1\).
\end{enumerate}
The following modified construction yields \textit{asymmetric bicycle (AB) codes}: \(\mathcal{Q}'=\text{CSS}(H'_{X},H_{Z})\) has \(H'_{X}=(A_{1}|B_{1}),A_{1}=a_{1}(P),B_{1}=b{1}(P)\), where \(a_{1}=\frac{a(x)}{\text{gcd}(a,b)},b_{1}=\frac{b(x)}{\text{gcd}(a,b)}\).
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Given the parameters \([n_{0}=2\ell,k_{0},d_{0}]_q\) of the classical linear quasi-cyclic code QC\((a,b)\), the quantum CSS code GB\((a,b)\) has parameters \([[ 2\ell,2k_{0}-2\ell,d]]_q\) where \(d\geq d_{0}\).
Consider a quasi-cyclic QC\((a,b)\) in the special case \(a(x)=f(x)h(x),b(x)=h(x)\), where for some polynomial \(r(x)\), \(\text{gcd}(a(x),b(x),x^{\ell}-1)=p(x)\) is a factor of the generating polynomial, \(g(x)=p(x)q(x)\). Then the distance of the QC code satisfies the following two bounds:
(a) If \(r(x)=0,d_{0}\geq\text{min}{d[q],1+d[q]}\);
(b) Otherwise, if gcd\((a(x),b(x),x^{\ell}-1)=1\), then \(d_{0}\text{min}{2d[q],d[q]/\text{wgt}(r)}\).
\begin{enumerate}[(a)]
\item If \(r(x)=0,d_{0}\geq\text{min}\{d[q],1+d[q]\}\);
\item Otherwise, if gcd\((a(x),b(x),x^{\ell}-1)=1\), then \(d_{0}\text{min}{2d[q],d[q]/\text{wgt}(r)}\).
\end{enumerate}
Here, \(h(x)=\text{gcd}(a(x),b(x),x^{\ell}-1)\) and \(g(x)=\frac{x^{\ell}-1}{h(x)}\), and \(d[q]\) is the distance of the linear cyclic code generated by \(q(x)\) \cite{arxiv:2203.17216}.
Let \(x^{\ell}-1=g(x)h(x)\) with \(g(x)\in \mathbb{F}_q[x]\) irreducible, and
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