Generalized bicycle (GB) code[1–3] 

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 \(d_{0}\text{min}{2d[q],d[q]/\text{wgt}(r)}\).

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 \(q(x)\) [3].

Let \(x^{\ell}-1=g(x)h(x)\) with \(g(x)\in \mathbb{F}_q[x]\) irreducible, and \begin{align} \label{eq:gb_gv} d_{GV}=\text{max }d:\sum_{s=1}^{d-1}(q-1)^{s}\left[\begin{pmatrix}2\ell\\ s \end{pmatrix}-\begin{pmatrix}\ell\\ s \end{pmatrix}\right]\,. \tag*{(3)}\end{align} Then, there exists \(f(x)\in \mathbb{F}_q[x]\) such that the length-\(2\ell\) quasi-cyclic code QC\((hf,h)\) has distance \(d\geq\text{min}(d[g],d_{GV})\), where \(d[g]\) is the distance of the cyclic code generated by \(g(x)\) [3].

GB codes with linear distance

Let \(\ell\) be such that \(\text{ord}_{\ell}-1=\ell-1\), where \(\text{ord}_{\ell}(q)\) is the multiplicative order function of \(q\) modulo \(\ell\). This ensures that \(x^{\ell}-1\) has only two irreducible factors in \(\mathbb{F}_q[x]\), \(h(x)=1-x\), and \(g(x)=1+x+\cdots+x^{\ell-1}\). Then, there is a GB code with parameters \([[ 2\ell,2,d\geq d_{GV}]]_q\) [3].

An incommensurate GB code with row weight \(w\) and parameters \([[ n=2\ell,k,d]]_p\) is equivalent to a CSS code local in \(D\leq w-1\) dimensions (\(D\leq w-1\) if \(\ell\) is prime). Its parameters satisfy the inequalities \(d\leq\mathcal{O}(n^{1-1/D})\) and \(kd^{2/(D-1)}\leq\mathcal{O}(n)\) [3].

A weight-four GB code of an odd distance \(d=2r+1\) must have length \(n\geq 1+d^{2}\). For an even distance \(d=2r\), the length \(n\geq d^{2}\) [3].