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codes/quantum/qudits_galois/qldpc/algebraic/generalized_bicycle.yml

@@ -77,21 +77,23 @@ features:
 relations:
     parents:
       - code_id: 2bga
-        detail: 'A code GB\((a,b)\) with circulants of size \(\ell\) is a special case of a 2BGA code over the cyclic group \(\mathbb{Z}_{\ell}\).
-        More precisely, for the cyclic group \(\mathbb{Z}_{\ell}\equiv \langle x|x^\ell=1\rangle \),
-        any element \(a\) of the \hyperref[topic:group-algebra]{group algebra} \(\mathbb{F}_q[\mathbb{Z}_{\ell}]\) can be seen as a
-        polynomial \(a(x)\in \mathbb{F}_q[x]\) over the group generator \(x\), where the polynomial degree deg\(a(x)<\ell\).
-        The 2BGA code LP\((a,b)\) is then just a generalized bicycle code GB\([a(x),b(x)]\) constructed from the polynomials \(a(x)\) and \(b(x)\) corresponding to \(a,b\in \mathbb{F}_q[\mathbb{Z}_{\ell}]\).'
+        detail: |
+          A code GB\((a,b)\) with circulants of size \(\ell\) is a special case of a 2BGA code over the cyclic group \(\mathbb{Z}_{\ell}\).
+          More precisely, for the cyclic group \(\mathbb{Z}_{\ell}\equiv \langle x|x^\ell=1\rangle \), any element \(a\) of the \hyperref[topic:group-algebra]{group algebra} \(\mathbb{F}_q[\mathbb{Z}_{\ell}]\) can be seen as a         polynomial \(a(x)\in \mathbb{F}_q[x]\) over the group generator \(x\), where the polynomial degree deg\(a(x)<\ell\).
+          The 2BGA code LP\((a,b)\) is then just a generalized bicycle code GB\([a(x),b(x)]\) constructed from the polynomials \(a(x)\) and \(b(x)\) corresponding to \(a,b\in \mathbb{F}_q[\mathbb{Z}_{\ell}]\).
       - code_id: lifted_product
-        detail: 'A code GB\((a,b)\) with circulants of size \(\ell\) is a special (degenerate) case of a lifted-product code LP\((A,B)\) code over the Abelian \hyperref[topic:group-algebra]{group algebra} \(\mathbb{F}_q[\mathbb{Z}_{\ell}]\) associated with the cyclic group \(\mathbb{Z}_{\ell}\equiv \langle x|x^\ell=1\rangle\), with \(1\times 1\) matrices \(A=a(x)\), \(B=b(b)\) given by the corresponding polynomials.'
+        detail: |
+          A code GB\((a,b)\) with circulants of size \(\ell\) is a special (degenerate) case of a lifted-product code LP\((A,B)\) code over the Abelian \hyperref[topic:group-algebra]{group algebra} \(\mathbb{F}_q[\mathbb{Z}_{\ell}]\) associated with the cyclic group \(\mathbb{Z}_{\ell}\equiv \langle x|x^\ell=1\rangle\), with \(1\times 1\) matrices \(A=a(x)\), \(B=b(b)\) given by the corresponding polynomials.
       - code_id: quantum_quasi_cyclic
-        detail: 'An index-\(m\) quasi-cyclic (QC) code of length \(n=m\ell\) is usually defined as a linear code invariant under the \(m\)-step shift permutation \(T_{n}^{m}\).'
+        detail: |
+          An index-\(m\) quasi-cyclic (QC) code of length \(n=m\ell\) is usually defined as a linear code invariant under the \(m\)-step shift permutation \(T_{n}^{m}\).
     cousins:
       - code_id: sc_qldpc
         detail: 'Qubit GB stabilizer generator matrices is equivalent to a 1D SC-QLDPC code, see \cite[Remark 7]{arxiv:2305.00137}.'
       - code_id: qldpc
-        detail: 'Stabilizer generators of the code GB\((a,b)\) have weights given by the sum of weights of polynomials \(a(x)\) and \(b(x)\).
-        The GB code ansatz is convenient for designing quantum LDPC codes.'
+        detail: |
+          Stabilizer generators of the code GB\((a,b)\) have weights given by the sum of weights of polynomials \(a(x)\) and \(b(x)\).
+          The GB code ansatz is convenient for designing quantum LDPC codes.
       - code_id: single_shot
         detail: 'A qubit GB code \([[n,k,d]]_2\) has \(k\) non-trivial relations between the syndrome bits, which is expected to help with operation in a fault-tolerant regime (in the presence of syndrome measurement errors). See Ref. \cite{arXiv:2306.16400} for many examples of such codes.'
       - code_id: quantum_cyclic