~

codes/classical/bits/cyclic/extended_golay.yml

@@ -35,6 +35,8 @@ relations:
       detail: 'The extended Golay code is nearly perfect.'
     - code_id: self_dual
       detail: 'The extended Golay code is the unique code at its parameters and happens to be self-dual \cite{doi:10.1016/0012-365X(75)90047-3}\cite[Remark 4.3.11]{preset:HPRainsSloane}.'
+    - code_id: binary_quad_residue
+      detail: 'The extended Golay code is an extended binary quadratic-residue code \cite[Ch. 16]{preset:MacSlo}.'
     - code_id: orthogonal_array
       detail: 'The extended Golay code is an orthogonal array of strength 7 \cite[Exam. 1]{doi:10.1109/18.720545}.'
     - code_id: group

codes/classical/bits/cyclic/self_dual_48_24_12.yml

@@ -17,6 +17,7 @@ description: |
 relations:
   parents:
     - code_id: binary_quad_residue
+      detail: 'The \([48,24,12]\) self-dual code is an extended quadratic-residue code \cite[Ch. 16]{preset:MacSlo}.'
     - code_id: self_dual
       detail: 'The \([48,24,12]\) self-dual code is the only self-dual doubly even code at its parameters \cite{doi:10.1109/TIT.2002.806146}.'
     - code_id: divisible

codes/classical/bits/easy/hamming/extended_hamming.yml

@@ -16,6 +16,10 @@ description: |
 relations:
   parents:
     - code_id: binary_linear
+    - code_id: quasi_cyclic
+      detail: 'The extended Hamming code is equivalent to a double circulant code \cite[pg. 497]{preset:MacSlo}.'
+    - code_id: binary_quad_residue
+      detail: 'The extended Hamming code is an extended binary quadratic-residue code \cite[Ch. 16]{preset:MacSlo}.'
     - code_id: small_distance
   cousins:
     - code_id: univ_opt_q-ary

codes/classical/bits/easy/hamming/hamming743.yml

@@ -36,7 +36,7 @@ relations:
   parents:
     - code_id: hamming
     - code_id: binary_quad_residue
-      detail: '\([7,4,3]\) Hamming code is a quadratic-residue code with generator polynomial \(1+x+x^3\) \cite{preset:MacSlo}.'
+      detail: 'The \([7,4,3]\) Hamming code is a quadratic-residue code with generator polynomial \(1+x+x^3\) \cite{preset:MacSlo}.'
   cousins:
     - code_id: incidence_matrix
       detail: 'The \([7,4,3]\) Hamming code parity-check matrix corresponds to points in the Fano plane \(PG_2(2)\) \cite[Exam. 21.4.2]{preset:HKSgraphs}.'

codes/classical/bits/easy/hamming/hamming844.yml

@@ -21,6 +21,8 @@ relations:
       detail: 'The \([8,4,4]\) extended Hamming code is a first-order RM code because it is self-dual and first-order RM codes are dual to extended Hamming codes.'
     - code_id: self_dual
       detail: 'The \([8,4,4]\) extended Hamming code is the smallest doubly even self-dual code.'
+    - code_id: binary_quad_residue
+      detail: 'The \([8,4,4]\) extended Hamming code is an extended quadratic-residue code \cite{preset:MacSlo}.'
   cousins:
     - code_id: divisible
       detail: 'The \([8,4,4]\) extended Hamming code code is the smallest double-even self-dual code.'

codes/classical/groups/permutation/rank_modulation.yml

@@ -10,7 +10,7 @@ name: 'Rank-modulation code'
 introduced: '\cite{doi:10.1109/ISIT.2008.4595285,arxiv:1110.2557}'
 
 description: |
-  A family of codes in permutations derived from \(q\)-ary linear codes, such as Lee-metric codes, RS codes \cite{arxiv:1110.2557}, quadratic residue codes, and most binary codes.
+  A family of codes in permutations derived from \(q\)-ary linear codes, such as Lee-metric codes, RS codes \cite{arxiv:1110.2557}, quadratic-residue codes, and most binary codes.
 
 features:
   rate: 'Rank modulation codes with code distance of \hyperref[topic:asymptotics]{order} \(d=\Theta(n^{1+\epsilon})\) for \(\epsilon\in[0,1]\) achieve a rate of \(1-\epsilon\) \cite{doi:10.1109/ISIT.2010.5513604}.'

codes/classical/properties/block/symmetry/cyclic/quasi_cyclic.yml

@@ -12,14 +12,15 @@ description: |
   A block code of length \(n\) is quasi-cyclic if, for each codeword \(c_1 \cdots c_{\ell} c_{\ell+1} \cdots c_n\), the string \(c_{n-\ell+1} \cdots c_n c_1 \cdots c_{n-\ell}\), where each entry is cyclically shifted by \(\ell\) increments, is also a codeword.
   Code for which \(\ell = 1\) are cyclic, while codes for which \(\ell = 2\) are called \textit{double circulant}.
 
-  The generator of an \([mn_0,mk_0]\) quasi-cyclic linear code is representable as a block matrix of \(m \times m\) circulant matrices \cite{manual:{Thomas A. Gulliver, \href{https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.143.3623}{Construction of quasi-cyclic codes}, Thesis, University of New Brunswick, 1989.}}.
+  The generator of an \([mn_0,mk_0]\) quasi-cyclic linear code is representable as a block matrix of \(m \times m\) circulant matrices \cite{preset:MacSlo,manual:{Thomas A. Gulliver, \href{https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.143.3623}{Construction of quasi-cyclic codes}, Thesis, University of New Brunswick, 1989.}}.
 
   Quasi-cyclic codes can also be understood in terms of the number of automorphism-group orbits required to generate all codewords.
   All codewords of a cyclic code can be obtained from any codeword via cyclic shifts, meaning that the code consists of only one orbit.
   On the other hand, quasi-cyclic codes consist of multiple disjoint orbits, meaning that not all of their codewords can be obtained from each other.
 
 notes:
   - 'A database of quasi-cyclic codes with searchable parameters such as block length and dimension is constructed and displayed \href{https://www.tec.hkr.se/~chen/research/codes/qc.htm}{here}.'
+  - 'See \cite[Ch. 16]{preset:MacSlo} for a review of double circulant codes.'
 
 relations:
   parents:

codes/classical/q-ary_digits/easy/hexacode.yml

@@ -41,8 +41,8 @@ relations:
     - code_id: evaluation
       detail: 'The hexacode is an evaluation AG code over the \hyperref[topic:finite-fields]{quaternary Galois field} \(GF(4) = \{0,1,\omega, \bar{\omega}\}\) with \(\cal X\) defined by \(x^2 y + \omega y^2 z + \bar{\omega} z^2 x = 0\) \cite[Exam. 2.77]{preset:HPAlgCodes}.'
     - code_id: q-ary_quad_residue
-      detail: 'The hexacode is the smallest example of an extended quadratic residue code of Type \(4^H\) \cite[Sec. 2.4.6]{doi:10.1007/3-540-30731-1}\cite[Exer. 363]{doi:10.1017/CBO9780511807077}.
-      The shortened hexacode is an odd-like quadratic residue code \cite[Exam. 6.6.8]{doi:10.1017/CBO9780511807077}.'
+      detail: 'The hexacode is the smallest example of an extended quadratic-residue code of Type \(4^H\) \cite[Sec. 2.4.6]{doi:10.1007/3-540-30731-1}\cite[Exer. 363]{doi:10.1017/CBO9780511807077}.
+      The shortened hexacode is an odd-like quadratic-residue code \cite[Exam. 6.6.8]{doi:10.1017/CBO9780511807077}.'
     - code_id: self_dual
       detail: 'The hexacode is Hermitian self-dual and, as a result, is also trace-Hermitian self-dual additive \cite[Sec. 9.10]{doi:10.1017/CBO9780511807077}.
       The hexacode and the shortened hexacode are extremal \cite[Tab. 9.14]{doi:10.1017/CBO9780511807077}\cite[Tm. 12]{arxiv:math/0005266}.'

codes/classical/q-ary_digits/easy/ternary_golay.yml

@@ -31,7 +31,7 @@ description: |
 
 #  The ternary golay code is a perfect code because with $\(d = 5\)$, \(\mathcal{G}_11\) can correct at most up to 2 errors so the Hamming bound
 #  is \( 3^11/\sum_{k=0}^{2}{n\choose k}2^k = 729\) which is exactly the number of codewords in \(\mathcal{G}_{11}\). In fact it can be shown that any non-trivial linear
-#  perfect code is either a Hamming code or a ternary or binary Golay code. A Golay code is also a quadratic residue code with residue set \(Q = \{1, 3, 4, 5, 9\} \) with
+#  perfect code is either a Hamming code or a ternary or binary Golay code. A Golay code is also a quadratic-residue code with residue set \(Q = \{1, 3, 4, 5, 9\} \) with
 #  generator polynomial \(x^5 + x^4 - x^3 + x^2 - 1\) over \(\mathbf{F}_3\).
 
 #protection: 'Corrects up to 2 errors and detects up to 4 errors'
@@ -52,7 +52,7 @@ realizations:
 relations:
   parents:
     - code_id: q-ary_quad_residue
-      detail: 'The ternary Golay code is a quadratic residue code over \(GF(3)\) with residue set \(Q = \{1, 3, 4, 5, 9\} \) and generator polynomial \(x^5 + x^4 - x^3 + x^2 - 1\) (\cite{preset:MacSlo}, Ch. 16).'
+      detail: 'The ternary Golay code is a quadratic-residue code over \(GF(3)\) with residue set \(Q = \{1, 3, 4, 5, 9\} \) and generator polynomial \(x^5 + x^4 - x^3 + x^2 - 1\) (\cite{preset:MacSlo}, Ch. 16).'
     - code_id: perfect
       detail: 'The ternary Golay code is perfect.'
     - code_id: delsarte_optimal_q-ary

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

@@ -12,13 +12,13 @@ short_name: '\(q\)-ary QR'
 #introduced: ''
 
 description: |
-  Member of a quadruple of cyclic \(q\)-ary codes of prime length \(n\) where \(q\) is prime and a quadratic residue modulo \(n\).
+  Member of a quadruple of cyclic \(q\)-ary codes of prime length \(n\) where \(q\) is prime and a quadratic-residue modulo \(n\).
   The codes are constructed using quadratic residues and nonresidues of \(n\).
   Extensions to prime-power \(q\) are also known \cite{doi:10.1109/TIT.1978.1055965,doi:10.1007/978-3-662-39641-4_6}.
 
   The roots of the generator polynomial \(r(x)\) of the first code (see \ref{topic:Cyclic-to-polynomial-correspondence}) are all of the inequivalent quadratic residues of \(n\), and the second code's generator polynomial is \((x-1)r(x)\). The roots of the generator polynomial \(a(x)\) of the third code are all inequivalent nonresidues of \(n\), and the fourth code's generator polynomial is \((x-1)a(x)\). The codes corresponding to polynomials \(r,a\) are often called \textit{augmented} quadratic-residue codes, while the remaining codes are called \textit{expurgated}.
 
-  The automorphism group of extended odd-like quadratic residue codes is \(PSL(2,GF(q))\), and these codes are the only codes with such symmetries \cite{arxiv:1704.01199}.
+  The automorphism group of extended odd-like quadratic-residue codes is \(PSL(2,GF(q))\), and these codes are the only codes with such symmetries \cite{arxiv:1704.01199}.
 
 features:
   rate: 'Achieve capacity of the binary erasure channel; see Ref. \cite{arxiv:2010.15453}.'

codes/classical/q-ary_digits/uplusv.yml

@@ -22,7 +22,7 @@ relations:
     - code_id: hamming743
       detail: 'Starting with the \([6,3,3]\) shortened Hamming code and applying the \((u|u+v)\) construction recursively using the repetition code yields a family of \([2^m,m+1,2^{m-1}]\) codes for \(m\geq1\) that saturate the Griesmer bound \cite[pg. 90]{doi:10.1201/9781315371993}.'
     - code_id: binary_quad_residue
-      detail: 'The \((u|u+v)\) construction can be used to obtain nonlinear binary quadratic residue codes \cite{doi:10.1109/TIT.1970.1054540}.'
+      detail: 'The \((u|u+v)\) construction can be used to obtain nonlinear binary quadratic-residue codes \cite{doi:10.1109/TIT.1970.1054540}.'
 
 
 # Begin Entry Meta Information

codes/quantum/qudits_galois/stabilizer/qldpc/balanced_product/lp/scalar/generalized_bicycle.yml

@@ -80,7 +80,7 @@ protection: |
 features:
   rate: |
     GB codes can achieve an asymptotic rate of  1/4 \cite{arxiv:2203.17216}.
-    For an odd prime \(\ell\), let a prime \(p\) be a quadratic residue modulo \(\ell\), i.e. \(p=m^{2}\text{mod}\ell\) for some integer \(m\).
+    For an odd prime \(\ell\), let a prime \(p\) be a quadratic-residue modulo \(\ell\), i.e. \(p=m^{2}\text{mod}\ell\) for some integer \(m\).
     Then, \(x^{\ell}-1\) has only three irreducible factors in \(\mathbb{F}_q(x)\), and there is a quadratic-residue cyclic code \([\ell,(\ell+1)/2, d]_p\) with \(d\geq\sqrt{\ell}\) and an irreducible generator polynomial.
     Using the GV distance \(d_{GV}\), a prime-field GB code with parameters \([[ 2\ell,(\ell-1)/2,d\geq \ell^{1/2}]]_p\) exists.
   decoders: