~

codes/classical/bits/reed_muller.yml

@@ -51,7 +51,7 @@ relations:
     - code_id: divisible
       detail: 'An RM\((r,m)\) code is \(2^{\left\lceil m/r\right\rceil-1}\)-divisible, according to McEliece''s theorem \cite{doi:10.1016/0097-3165(71)90066-5,doi:10.1016/0012-365X(72)90032-5}.'
     - code_id: group
-      detail: 'RM codes are group-algebra codes \cite{doi:10.1007/BF01072842,manual:{Charpin, Pascale. Codes idéaux de certaines algèbres modulaires. Diss. 1982.}}\cite[Ex. 16.4.11]{preset:HKSalgebra}. Consider a binary vector space of dimension \( m \). Under addition, this forms a finite group with \( 2^m \) elements known as an elementary abelian 2-group -- the direct product of \( m \) two-element cyclic groups \( \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2 \). Denote this group by \( G_m \). Let \( J \) be the Jacobson radical of the group algebra \( \mathbb{F}_2 G_m \), where \(\mathbb{F}_2=GF(2)\). RM\((r,m)\) codes correspond to the ideal \( J^{m-r} \). The length of the code is \( |G_m| = 2^m \), the distance is \( 2^{m-r} \), and the dimension is \( \sum_{i=0}^r {m \choose i} \). A similar construction exists for choices of a prime \( p\neq 2 \).'
+      detail: 'RM codes are group-algebra codes \cite{doi:10.1007/BF01072842,manual:{Charpin, Pascale. Codes idéaux de certaines algèbres modulaires. Diss. 1982.}}\cite[Ex. 16.4.11]{preset:HKSalgebra}. Consider a binary vector space of dimension \( m \). Under addition, this forms a finite group with \( 2^m \) elements known as an elementary abelian 2-group -- the direct product of \( m \) two-element cyclic groups \( \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2 \). Denote this group by \( G_m \). Let \( J \) be the Jacobson radical of the \hyperref[topic:group-algebra]{group algebra} \( \mathbb{F}_2 G_m \), where \(\mathbb{F}_2=GF(2)\). RM\((r,m)\) codes correspond to the ideal \( J^{m-r} \). The length of the code is \( |G_m| = 2^m \), the distance is \( 2^{m-r} \), and the dimension is \( \sum_{i=0}^r {m \choose i} \). A similar construction exists for choices of a prime \( p\neq 2 \).'
   cousins:
     - code_id: bch
       detail: 'RM\(^*(r,m)\) codes are equivalent to subcodes of BCH codes of designed distance \(2^{m-r}-1\) while RM\((r,m)\) are subcodes of extended BCH codes of the same designed distance \cite[Ch. 13]{preset:MacSlo}.'

codes/classical/q-ary_digits/cyclic/group.yml

@@ -9,13 +9,32 @@ logical: q-ary_digits
 
 name: 'Group-algebra code'
 #introduced: ''
+alternative_names:
+  - 'Group code'
 
 description: |
-  Also known as a \textit{group code}.
   An \( [n,k]_q \) code based on a finite group \( G \) of size \(n \).
   A group-algebra code for an abelian group is called an \textit{abelian group-algebra code}.
 
-  The code is a \( k \)-dimensional linear subspace of the group algebra of \( G\) with coefficients in the field \(GF(q) = \mathbb{F}_q\) with \(q\) elements. To be precise, the code must be closed under permutations corresponding to the elements of the group \( G \); therefore, \( G \) must be a subgroup of the permutation automorphism group of the code, which is defined as the group of permutations of the physical bits that preserve the code space. This leads us to the formal definition of a group-algebra code: a group-algebra code is an ideal in the group algebra \( \mathbb{F}_q G \).
+  \subsection{Group algebra}
+  \label{topic:group-algebra}
+  For a given field \(\mathbb{F}_q\) and a finite group \(G\) of order
+  \(|G|=\ell\), the \textit{group algebra} (a ring) \(\mathbb{F}_q[G]\) is defined as an
+  \(\mathbb{F}_q\)-linear space of all formal sums
+  \begin{align}
+    \label{eq:algebra-element}
+    x\equiv \sum_{g\in G}x_g g,\quad x_g\in \mathbb{F}_q,
+  \end{align}
+  where group elements \(g\in G\) serve as basis vectors,
+  equipped with the product naturally associated with the group
+  operation,
+  \begin{align}
+    \label{eq:FG-product}
+    ab=\sum_{g\in G}\biggl(\sum_{h\in G} a_h b_{h^{-1}g}\biggr) g, \quad a,b\in \mathbb{F}_q[G].
+  \end{align}
+
+  \subsection{Group-algebra code}
+  A group-algebra code is a \( k \)-dimensional linear subspace of the \hyperref[topic:group-algebra]{group algebra} of \( G\) with coefficients in the field \(GF(q) = \mathbb{F}_q\) with \(q\) elements. To be precise, the code must be closed under permutations corresponding to the elements of the group \( G \); therefore, \( G \) must be a subgroup of the permutation automorphism group of the code, which is defined as the group of permutations of the physical bits that preserve the code space. This leads us to the formal definition of a group-algebra code: a group-algebra code is an ideal in the \hyperref[topic:group-algebra]{group algebra} \( \mathbb{F}_q G \).
 
 #protection: 'The class of abelian group-algebra codes is very general, for example including all group-algebra codes of size \(n \leq 23 \). As such it is very difficult to say anything about the distance of abelian groups codes without specializing to a particular family'
 

codes/quantum/qubits/stabilizer/qldpc/algebraic/qcga.yml

@@ -7,26 +7,29 @@ code_id: qcga
 physical: qubits
 logical: qubits
 
-name: 'High-rate group-algebra code'
+name: 'Bravyi-Cross-Gambetta-Maslov-Rall-Yoder (BCGMRY) group-algebra code'
+short_name: 'BCGMRY code'
 introduced: '\cite{arxiv:2308.07915}'
 
 description: |
-  One of several specific 2BGA QLDPC qubit codes over Abelian groups with high encoding rate but with minimal non-geometrically local qubit connectivity.
+  One of several Abelian 2BGA codes which admit time-optimal syndrome measurement circuits that can be implemented in a two-layer
+  architecture, a generalization of the square-lattice architecture
+  optimal for the surface codes.
 
   The qubit connectivity graph is not quite a 2D grid and is instead decomposable into two planar subgraphs of degree three.
   There are \(n\) \(X\) and \(Z\) check operators, with each one of weight six.
 
 features:
   rate: 'When ancilla qubit overhead is included, the encoding rate surpasses that of the surface code.
-  A general \([[n,k,d]]\) high-rate group-algebra code requires \(n\) ancilla qubits for encoding, meaning that its \textit{ancilla-added encoding rate} is \(k/2n\).
+  A general \([[n,k,d]]\) BCGMRY code requires \(n\) ancilla qubits for encoding, meaning that its \textit{ancilla-added encoding rate} is \(k/2n\).
   For example, the \([[144,12,12]]\) code has ancilla-added rate \(1/24\).
   In contrast, the distance-13 surface code has ancilla-added rate \(1/338\).'
 
   fault_tolerance:
     - 'Fault-tolerant state initialization using Tanner graph techniques \cite{arxiv:2110.10794} and an ancillary surface code \cite{arxiv:2308.07915}.'
 
   decoders:
-    - 'Syndrome extraction circuit requires seven layers of CNOT gates regardless of code length. BP-OSD decoder \cite{arxiv:1904.02703} has been extended to account for measurement errors (i.e., the circuit-based noise model \cite{arxiv:0803.0272}) \cite{arxiv:2308.07915}.'
+    - 'Syndrome extraction circuit requires seven layers of CNOT gates regardless of code length. BP-OSD decoder \cite{arxiv:1904.02703} has been extended \cite{arxiv:2308.07915} to account for measurement errors (i.e., the circuit-based noise model \cite{arxiv:0803.0272}).'
 
   threshold:
     - '\(0.8\%\) for circuit-level noise under BP-OSD decoder \cite{arxiv:2308.07915} (cf. \cite{arxiv:0803.0272}).'
@@ -36,11 +39,11 @@ relations:
     parents:
       - code_id: qubit_css
       - code_id: 2bga
-        detail: 'A high-rate group-algebra code is a special case of a 2BGA code over an Abelian group \(G\).'
+        detail: 'BCGMRY codes are Abelian 2BGA codes with stabilizer generators of weight 6.'
       - code_id: qldpc
     cousins:
       - code_id: surface
-        detail: 'High-rate group-algebra codes are on par with the surface code in terms of threshold, but admit a much higher ancilla-added encoding rate at the expense of having non-geometrically local weight-six check operators.'
+        detail: 'BCGMRY codes are on par with the surface code in terms of threshold, but admit a much higher ancilla-added encoding rate at the expense of having non-geometrically local weight-six check operators.'
 
 
 # Begin Entry Meta Information

codes/quantum/qudits_galois/qldpc/algebraic/2bga.yml

@@ -13,25 +13,11 @@ introduced: '\cite{arXiv:2306.16400}'
 
 description: |
   2BGA codes are the smallest \hyperref[code:lifted_product]{LP codes}
-  LP\((a,b)\), constructed from a pair of group algebra elements
-  \(a,b\in \mathbb{F}_q[G]\), where \(G\) is a finite group and \(\mathbb{F}_q\) is a Galois field.
+  LP\((a,b)\), 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
   \(n=2\ell\).
-
-  For a given field \(\mathbb{F}_q\) and a finite group \(G\) of order
-  \(|G|=\ell\), the \textit{group algebra} (a ring) \(\mathbb{F}_q[G]\) is defined as an
-  \(\mathbb{F}_q\)-linear space of all formal sums
-  \begin{align}
-    \label{eq:algebra-element}
-    x\equiv \sum_{g\in G}x_g g,\quad x_g\in \mathbb{F}_q,
-  \end{align}
-  where group elements \(g\in G\) serve as basis vectors,
-  equipped with the product naturally associated with the group
-  operation,
-  \begin{align}
-    \label{eq:FG-product}
-    ab=\sum_{g\in G}\biggl(\sum_{h\in G} a_h b_{h^{-1}g}\biggr) g, \quad a,b\in \mathbb{F}_q[G].
-  \end{align}
+  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
   naturally defined as a linear space of pairs
@@ -43,68 +29,78 @@ description: |
   and \(v'=v-aw\) identified.  The order in the products is relevant
   when the group is non-Abelian.
 
-  \subsection{Examples}
-
-  1. GB codes: For the cyclic group \(\mathbb{Z}_{\ell}\equiv \langle x|x^\ell=1\rangle \),
-  any element \(a\) of the 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 just
-  a generalized bicycle code GB\([a(x),b(x)]\) constructed from the
-  polynomials \(a(x)\) and \(b(x)\) corresponding to
-  \(a,b\in \mathbb{F}_q[\mathbb{Z}_{\ell}]\).
-
-  2. Multi-dimensional quantum quasi-cyclic codes: Any finite abelian
-  group can be written as a direct product of several cyclic groups,
-  e.g., \(G=C_{m_1}\times C_{m_2}\times \ldots C_{m_D}\) for a product
-  of \(D\) cyclic groups, which is equivalent to a representation
-  \begin{align}
-    G=\langle x_1,\ldots,x_D|x_1^{m_1}=x_2^{m_2}=\ldots=x_D^{m_D}\rangle.
-  \end{align}
-  Respectively, an element of the group algebra \(\mathbb{F}_q[G]\), where \(\mathbb{F}_q\) is
-  a finite field, can be written as a \(D\)-variate polynomial in
-  \(\mathbb{F}_q[x_1,x_2,\ldots,x_D]\), with the degree of the generator \(x_j\) of
-  order \(m_j\) not exceeding \(m_j-1\).  Thus, any abelian 2BGA code
-  can be thought of as a multi-dimensional index-two quasi-cyclic code.
-  This includes toric codes and Haah's cubic codes.  An equivalent
-  construction in terms of Kronecker products of circulant matrices was
-  introduced in \cite{arXiv:1212.6703}.
-  Related higher-dimensional
-  quasi-cyclic and convolutional quantum codes have been constructed in
-  \cite{arxiv:2305.00137}.
-
-  3. Several abelian 2BGA codes with stabilizer generators of weight 6
-  have been constructed by Bravyi et al. \cite{arXiv:2308.07915}.  These
-  are two-dimensional QC codes which admit time-optimal syndrome
-  measurement circuits which can be implemented in a two-layer
-  architecture, a generalization of the square-lattice architecture
-  optimal for the surface codes.  Circuit simulations show a threshold
-  of approximately \(0.8\%\).
-
-  4. Explicit example from \cite{arXiv:2306.16400}: Consider the
-  alternating group \(A_4\), also known as the rotation group of a
+  For example, consider the
+  alternating group \(G=A_4=T\), also known as the rotation group of a
   regular tetrahedron,
   \begin{align}
     T=\langle x,y|x^3=(yx)^3=y^2=1\rangle,\quad |T|=12,
   \end{align} and the
-  binary algebra \(\mathbb{F}_q_2[T]\).  Select \(a=1+x+y+x^{-1}yx\)
-  and \(b=1+x+y+yx\) to get an \emph{essentially non-abelian} 2BGA code
-  LP\([a,b]\) with parameters \([[24,5,3]]_2\).
+  binary algebra \(\mathbb{F}_2[T]\).  Select \(a=1+x+y+x^{-1}yx\)
+  and \(b=1+x+y+yx\) to get an \emph{essentially non-Abelian} 2BGA code
+  LP\([a,b]\) with parameters \([[24,5,3]]_2\) \cite{arXiv:2306.16400}.
+
+# \subsection{Examples}
+#
+# 1. GB codes: 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 just
+# a generalized bicycle code GB\([a(x),b(x)]\) constructed from the
+# polynomials \(a(x)\) and \(b(x)\) corresponding to
+# \(a,b\in \mathbb{F}_q[\mathbb{Z}_{\ell}]\).
+#
+# 2. Multi-dimensional quantum quasi-cyclic codes: Any finite Abelian
+# group can be written as a direct product of several cyclic groups,
+# e.g., \(G=C_{m_1}\times C_{m_2}\times \ldots C_{m_D}\) for a product
+# of \(D\) cyclic groups, which is equivalent to a representation
+# \begin{align}
+#   G=\langle x_1,\ldots,x_D|x_1^{m_1}=x_2^{m_2}=\ldots=x_D^{m_D}\rangle.
+# \end{align}
+# Respectively, an element of the \hyperref[topic:group-algebra]{group algebra} \(\mathbb{F}_q[G]\), where \(\mathbb{F}_q\) is
+# a finite field, can be written as a \(D\)-variate polynomial in
+# \(\mathbb{F}_q[x_1,x_2,\ldots,x_D]\), with the degree of the generator \(x_j\) of
+# order \(m_j\) not exceeding \(m_j-1\).  Thus, any Abelian 2BGA code
+# can be thought of as a multi-dimensional index-two quasi-cyclic code.
+# This includes toric codes and Haah's cubic codes.  An equivalent
+# construction in terms of Kronecker products of circulant matrices was
+# introduced in \cite{arXiv:1212.6703}.
+# Related higher-dimensional
+# quasi-cyclic and convolutional quantum codes have been constructed in
+# \cite{arxiv:2305.00137}.
+#
+# 3. Several Abelian 2BGA codes with stabilizer generators of weight 6
+# have been constructed by Bravyi et al. \cite{arXiv:2308.07915}.  These
+# are two-dimensional QC codes which admit time-optimal syndrome
+# measurement circuits which can be implemented in a two-layer
+# architecture, a generalization of the square-lattice architecture
+# optimal for the surface codes.  Circuit simulations show a threshold
+# of approximately \(0.8\%\).
+#
+# 4. Explicit example from \cite{arXiv:2306.16400}: Consider the
+# alternating group \(A_4\), also known as the rotation group of a
+# regular tetrahedron,
+# \begin{align}
+#   T=\langle x,y|x^3=(yx)^3=y^2=1\rangle,\quad |T|=12,
+# \end{align} and the
+# binary algebra \(\mathbb{F}_q_2[T]\).  Select \(a=1+x+y+x^{-1}yx\)
+# and \(b=1+x+y+yx\) to get an \emph{essentially non-Abelian} 2BGA code
+# LP\([a,b]\) with parameters \([[24,5,3]]_2\) \cite{arXiv:2306.16400}.
 
 protection: |
-  Some upper and lower bounds on parameters and many examples of 2BGA codes are given in \cite{arXiv:2306.16400}.
-  For a 2BGA code constructed from an abelian group, the code dimension \(k\) is always even \cite{manual:{Kalachev, G. V., and Panteleev, P. A. (2020). On the minimum distance in one class of quantum LDPC codes. Intelligent systems. Theory and applications, 24(4), 87-117.}}.
+  Some upper and lower bounds on parameters and many examples of 2BGA codes are given in Ref. \cite{arXiv:2306.16400}.
+  The code dimension \(k\) for Abelian 2BGA codes is always even \cite{manual:{Kalachev, G. V., and Panteleev, P. A. (2020). On the minimum distance in one class of quantum LDPC codes. Intelligent systems. Theory and applications, 24(4), 87-117.}}.
 
 features:
   rate: |-
-    2BGA construction gives some of the best short codes with small stabilizer weights.
+    The 2BGA construction gives some of the best short codes with small stabilizer weights.
 
     A number of 2BGA codes \([[n,k,d]]_q\) with row weights
     \(W\le 8\), block lengths \(n\le 100\), and parameters such that
-    \(kd\ge n\) have been constructed in \cite{arxiv:2306.16400} by
-    exhaustive enumeration.  Examples include GB codes with parameters
-    \([[70,8,10]]_2\), \([[72,10,9]]_2\), 2BGA codes from abelian groups
+    \(kd\ge n\) have been constructed by
+    exhaustive enumeration \cite{arxiv:2306.16400}.  Examples include GB codes with parameters
+    \([[70,8,10]]_2\), \([[72,10,9]]_2\), Abelian 2BGA for groups
     \(\mathbb{Z}_{mh}=\mathbb{Z}_m\times \mathbb{Z}_2\) (index-4 QC codes) with parameters
-    \([[48,8,6]]_2\) and \([[56,8,7]]_2\), and non-abelian codes with
+    \([[48,8,6]]_2\) and \([[56,8,7]]_2\), and non-Abelian codes with
     parameters \([[64,8,8]]_2\), \([[82,10,9]]_2\), \([[96,10,12]]_2\),
     and \([[96,12,10]]_2\) (all of these have stabilizer generators of
     weight \(W=8\).)
@@ -113,15 +109,36 @@ relations:
   parents:
     - code_id: lifted_product
       detail: |-
-        2BGA codes are the smallest \hyperref[code:lifted_product]{LP codes} LP\((a,b)\), constructed from a pair of one-by-one matrices \(a,b\in \mathbb{F}_q[G]\) in a group algebra.
+        2BGA codes are LP\((a,b)\) codes, constructed from a pair of one-by-one matrices \(a,b\in \mathbb{F}_q[G]\) in a \hyperref[topic:group-algebra]{group algebra}.
     - code_id: two_block_quantum
       detail: |-
-        2BGA codes are a special case of \hyperref[code:two_block_quantum]{two-block quantum codes} where the commuting matrices are constructed with the help of a group algebra.
+        2BGA codes are two-block quantum codes whose commuting matrices are constructed with the help of a \hyperref[topic:group-algebra]{group algebra}.
   cousins:
     - code_id: qldpc
       detail: |
-        Given group algebra elements \(a,b\in \mathbb{F}_q[G]\) with weights \(W_a\) and \(W_b\) [number of non-zero terms in the expansion], the 2BGA code LP\((a,b)\) has stabilizer
+        Given \hyperref[topic:group-algebra]{group algebra} elements \(a,b\in \mathbb{F}_q[G]\) with weights \(W_a\) and \(W_b\) (i.e., number of non-zero terms in the expansion), the 2BGA code LP\((a,b)\) has stabilizer
         generators of uniform weight \(W_a+W_b\).
+    - code_id: quantum_quasi_cyclic
+      detail: |
+        Any Abelian 2BGA code
+        can be thought of as a multi-dimensional index-two quasi-cyclic code.
+        More precisely, any finite Abelian
+        group can be written as a direct product of several cyclic groups,
+        e.g., \(G=C_{m_1}\times C_{m_2}\times \ldots C_{m_D}\) for a product
+        of \(D\) cyclic groups, which is equivalent to a representation
+        \begin{align}
+          G=\langle x_1,\ldots,x_D|x_1^{m_1}=x_2^{m_2}=\ldots=x_D^{m_D}\rangle.
+        \end{align}
+        Respectively, an element of the \hyperref[topic:group-algebra]{group algebra} \(\mathbb{F}_q[G]\), where \(\mathbb{F}_q\) is
+        a finite field, can be written as a \(D\)-variate polynomial in
+        \(\mathbb{F}_q[x_1,x_2,\ldots,x_D]\), with the degree of the generator \(x_j\) of
+        order \(m_j\) not exceeding \(m_j-1\).
+        An equivalent
+        construction in terms of Kronecker products of circulant matrices was
+        introduced in \cite{arXiv:1212.6703}.
+        Related higher-dimensional quasi-cyclic and convolutional quantum codes have been constructed in
+        \cite{arxiv:2305.00137}.
+    - code_id: group
 
 
 # Begin Entry Meta Information