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codes/classical/bits/nonlinear/kerdock.yml

@@ -12,18 +12,16 @@ name: 'Kerdock code'
 introduced: '\cite{doi:10.1016/S0019-9958(72)90376-2}'
 
 description: |
-  Binary nonlinear \((2^m, 2^{2m}, 2^{m-1} - 2^{frac{m}{2}-1})\) code where \(m\) is even. The size of code book is \(|2^{m-1}||R(1,m)|\), where \(R(1,m)\) is the corresponding first order Reed Muller code.
+  Binary nonlinear \((2^m, 2^{2m}, 2^{m-1} - 2^{(m-2)/2})\) code where \(m\) is even. The size of code book is \(|2^{m-1}||R(1,m)|\), where \(R(1,m)\) is the corresponding first order Reed Muller code.
   This is twice the size of the largest possible linear code with the same length and distance.
   The relative minimum distance tends to \(\frac{1}{2}\) as \(m\) becomes large.
 
-  Let the matrices \(A = [a_{ij}]\) constitute the Kerdock set \(K\), i.e., a set of symmetric binary matrices with zero diagonal entries with the property that difference of distinct matrices in the set have full rank.
-  Boolean functions are of the form \(Q(X) + l(X) + b\) where \(Q(X) is \sum_{1 \leq i < j \leq m} a_{ij}X_{i}X_{j}\), \(l(X)\) is linear in \(m\) variables and \(b\) is in
-  \(GF(2)\).
-  Codewords are  formed as evaluations of these functions over \(GF(2)^{m}\). The variable \(X\) represents vectors constituting
-  \(GF(2)^{m}\).
+  Let the matrices \(A = [a_{ij}]\) constitute the Kerdock set \(K\), i.e., a set of symmetric binary matrices with zero diagonal entries with the property that differences of distinct matrices in the set have full rank.
+  Define Boolean functions of the form \(Q(X) + l(X) + b\), where \(Q(X) = \sum_{1 \leq i < j \leq m} a_{ij}X_{i}X_{j}\), the function \(l(X)\) is linear in \(m\) variables, and \(b\) is a bit.
+  Codewords are formed as evaluations of these functions over \(GF(2)^{m}\) in the variable \(X \in GF(2)^{m}\).
 
 features:
-  rate: 'The transmission rate is \(\frac{2m}{2^m}\) which tends to 0 as \(m\) becomes large, hence these codes are asymptotically poor.'
+  rate: 'The transmission rate is \(2m/2^m\) which tends to 0 as \(m\) becomes large, hence these codes are asymptotically poor.'
 
   decoders:
     - 'Soft decision decoding involves extending the Fast Hadamard Transform decoding algorithm for the binary first-order Reed-Muller code to Kerdock code \cite{doi:10.1109/18.312154}.'
@@ -33,6 +31,9 @@ features:
 realizations:
   - 'Digital fingerprinting: \(t\)-collision secure schemes can be designed to detect a pirated copy once \(t\) users have colluded \cite{doi:10.1109/TIT.2004.838106}.'
 
+notes:
+  - 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Kerdock Codes.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CKerdock.html}}.'
+
 
 relations:
   parents:
@@ -41,14 +42,14 @@ relations:
     - code_id: quaternary_over_z4
       detail: 'Kerdock codes can be seen, via the Gray map, as extended linear cyclic codes over \(\mathbb{Z}_4\), where the appended bit is a zero sum check \cite{doi:10.1109/18.312154,arxiv:math/0207208}.'
     - code_id: reed_muller
-      detail: 'Kerdock code is a subcode of second order RM Code \cite[pg. 457]{preset:MacSlo}.
-      It consists of a number of cosets of \(RM(2,m)\) created by quotienting with \(RM(1,m)\).'
+      detail: 'Kerdock code is a subcode of a second-order RM Code \cite[pg. 457]{preset:MacSlo}.
+      It consists of a number of cosets of RM\((2,m)\) created by quotienting with RM\((1,m)\).'
     - code_id: preparata
-      detail: 'Preparata codes are dual to Kerdock codes in the sense of McWilliams weight distribution. These well-known codes are in fact equivalent to a pair of mutually dual linear codes over \(\mathbb{Z}_4\).'
+      detail: 'Preparata codes are dual to Kerdock codes in the sense of the MacWilliams weight distribution \cite{doi:10.1109/TIT.1983.1056676}. However, the two codes are images of a pair of mutually dual linear codes over \(\mathbb{Z}_4\) under the Gray map \cite[Sec. 6.3]{doi:10.1201/9781315147901}.'
     - code_id: hamming
       detail: 'Kerdock codes can be obtained by Hensel-lifting Hamming codes to \(\mathbb{Z}_4\) \cite{doi:10.1109/18.312154}.'
     - code_id: qubit_stabilizer
-      detail: 'Kerdock codes can be used to form a subset of stabilizer states, and the corresponding Clifford-group automorphisms of this set form a particular group \cite{doi:10.1112/S0024611597000403} that is also a unitary two-design \cite{arxiv:1904.07842}.'
+      detail: 'Kerdock codes can be used to form a subset of stabilizer states, and the corresponding Clifford-group automorphisms of this set form a particular group \cite{doi:10.1112/S0024611597000403} that is a unitary two-design \cite{arxiv:1904.07842}.'
     - code_id: univ_opt_q-ary
       detail: 'Kerdock codes are asymptotically universally optimal \cite[Exam. 12.3.25]{doi:10.1201/9781315147901}.'
     - code_id: 24cell

codes/classical/bits/nonlinear/preparata.yml

@@ -15,10 +15,10 @@ description: |
   A nonlinear binary \((2^{m+1}-1, 2^{m+1}-2m-2, 5)\) code where \(m\) is odd.
   The size of this code is twice the size of the largest possible linear code with the same length and distance.
 
-  Codewords of extended Preparata codes can be thought of as characteristic functions of the tuple \((X,Y)\), where \(X, Y\) belong to \(\mathbb{F}_{2^m}\), along with the following properties: (i) \(|X| = |Y| = 2^{m}\), (ii) \(\sum_{x \in X} x = \sum_{y \in Y} y \), and (iii) \(\sum_{x \in X} x^{3} + (\sum_{x \in X} x)^{3}  = \sum_{y \in Y} y^{3} \).
+  Codewords of extended Preparata codes can be thought of as characteristic functions of the tuple \((X,Y)\), where \(X, Y\) belong to \(GF(2)^m\), along with the following properties: (i) \(|X| = |Y| = 2^{m}\), (ii) \(\sum_{x \in X} x = \sum_{y \in Y} y \), and (iii) \(\sum_{x \in X} x^{3} + (\sum_{x \in X} x)^{3}  = \sum_{y \in Y} y^{3} \).
   Preparata codes are obtained by leaving out the coordinate in the zeroth positions in the first half.
 
-protection: 'Preparata codes are a class of non linear double error correcting codes. Since the minimum distance is \(5\), preparata codes can correct \(\frac{5-1}{2} = 4\) errors.'
+# protection: 'Preparata codes are a class of non linear double error correcting codes. Since the minimum distance is \(5\), preparata codes can correct \(\frac{5-1}{2} = 4\) errors.'
 
 
 features:
@@ -33,14 +33,12 @@ relations:
   parents:
     - code_id: nearly_perfect
       detail: 'Preparata codes are uniformly packed, quasi-perfect \cite[pg. 475]{preset:MacSlo}, and nearly perfect.
-      For any word \(textbf{u}\) and Preparata codebook \(C\) with \(d(\textbf{u}, C) > 2\), we have that \(u\) has a distance 2 or 3 to exactly floor((2^{m+1}-1)/3) codewords.'
+      For any word \(u\) and Preparata codebook \(C\) with \(d(u, C) > 2\), we have that \(u\) has a distance 2 or 3 to exactly floor((2^{m+1}-1)/3) codewords.'
   cousins:
     - code_id: quaternary_over_z4
       detail: 'Preparata codes can be seen, via the Gray map, as linear codes over \(\mathbb{Z}_4\) \cite{doi:10.1109/18.312154,arxiv:math/0207208}.'
-    - code_id: kerdock
-      detail: 'Weight enumerator of Kerdock and Preparata codes are McWilliams Dual of each other. This has been regarded as a coincidence \cite{doi:10.1109/TIT.1983.1056676}. However, when thought of as linear codes over \(\mathbb{Z}_4\), the two codes are dual to each other \cite[Sec. 6.3]{doi:10.1201/9781315147901}.'
     - code_id: reed_muller
-      detail: 'A Preparata code can be written as a union of a linear subcode \(\mathcal{C}\) of RM\((m-2,m)\) and the \(2^{m-1}-1\) representatives of coset formed by \(\mathcal{C}\) in RM\((m-2,m)\). The coset representatives are given by \(|1|x^j|0|x^{j}\theta_{1}|\), where \(1\leq j \leq 2^{m-1}-1\). \(\mathcal{C}\) comprises of codewords of the form \(|g(1)|g(x)(1+\theta_{1})|f(1)+g(1)|g(x)(1+\theta_{1})+f(x)(1+\theta_{1}+\theta_{3})|\), where \(f(x)\) and \(g(x)\) are arbitrary, and where \(\theta_{1}\) and \(theta_{3}\) denote the primitive idempotents corresponding to cyclotomic cosets \(C_1\) and \(C_3\) respectively.'
+      detail: 'A Preparata code can be written as a union of a linear subcode \(\mathcal{C}\) of RM\((m-2,m)\) and the \(2^{m-1}-1\) representatives of coset formed by \(\mathcal{C}\) in RM\((m-2,m)\). The coset representatives are given by \(|1|x^j|0|x^{j}\theta_{1}|\), where \(1\leq j \leq 2^{m-1}-1\). \(\mathcal{C}\) comprises of codewords of the form \(|g(1)|g(x)(1+\theta_{1})|f(1)+g(1)|g(x)(1+\theta_{1})+f(x)(1+\theta_{1}+\theta_{3})|\), where \(f(x)\) and \(g(x)\) are arbitrary, and where \(\theta_{1}\) and \(\theta_{3}\) denote the primitive idempotents corresponding to cyclotomic cosets \(C_1\) and \(C_3\) respectively.'
     - code_id: bch
       detail: 'Preparata codes contain twice as many code words as the double-error-correcting BCH codes of the same length, which is the largest number of code words possible for given length and distance \cite{doi:10.1016/S0019-9958(68)90874-7}.'
     - code_id: hamming

codes/classical/spherical/q-ary/kerdock_spherical.yml

@@ -10,15 +10,15 @@ logical: reals
 name: 'Kerdock spherical code'
 
 description: |
-   Family of \((n=2^{2r},n^2,2-2/\sqrt{n})\) spherical codes for \(r \geq 2\) obtained from Kerdock codes via the binary antipodal mapping \cite[pg. 157]{preset:EricZin}.
+   Family of \((n=2^{2r},n^2,2-2/\sqrt{n})\) spherical codes for \(r \geq 2\), obtained from Kerdock codes via the binary antipodal mapping \cite[pg. 157]{preset:EricZin}.
 
 
 relations:
   parents:
     - code_id: binary_antipodal
   cousins:
     - code_id: kerdock
-      detail: 'Kerdock spherical codes can be obtained from Kerdock codes using the binary antipodal mapping.'
+      detail: 'Kerdock spherical codes can be obtained from Kerdock codes using the binary antipodal mapping \cite[pg. 157]{preset:EricZin}.'
     - code_id: univ_opt_spherical
       detail: 'Kerdock spherical codes are almost universally optimal \cite{arxiv:1509.07837}.'
 

codes/quantum/qudits_galois/evaluation/galois_reed_muller.yml

@@ -13,29 +13,30 @@ short_name: 'QRM'
 introduced: '\cite{arXiv:quant-ph/0502001}'
 
 description: |
-  True \(q\)-Galois-qudit stabilizer code constructed from generalized Reed-Solomon (GRS) codes via either the Hermitian construction or the Galois-qudit CSS construction.
+  True \(q\)-Galois-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes via either the Hermitian construction or the Galois-qudit CSS construction.
 
   The CSS construction yields a
-  \([[q^m,k(v_2)-k(v_1),min{d(v_1^{\perp}),d(v_2)}]]_q\) code, constructed from the generalized Reed Muller Codes
+  \([[q^m,k(v_2)-k(v_1),\min\{d(v_1^{\perp}),d(v_2)\}]]_q\) code, constructed from the generalized Reed Muller Codes
   RM\(_q(v_1,m)\) and RM\(_q(v_2,m)\), with \(0\leq v_1 \leq v_2 \leq q(m-1)-1\).
   The parameters are \(k(v) = \sum_{j=0}^{m}(-1)^{j}\dbinom{m}{j}\dbinom{m+v-jq}{v-jq}\) and
   \(d(v) = (R+1)q^{Q}\), where \(m(q-1)-v=(q-1)Q+R\) so that \(0\leq R\leq q-1\).
-  Here \(0\leq v_1,v_2 leq m(q-1)-1\), \(q\) is prime, and \(m\) is a positive integer.
+  Here \(0\leq v_1,v_2 \leq m(q-1)-1\), \(q\) is prime, and \(m\) is a positive integer.
 
-  Using the code GRM\(_{q^2}(v,m)\) for \(0\leq v \leq q(m-1)-1\), the Hermitian construction yields a pure \([[q^{2m},q^{2m}-k(v),d(v^{\perp}]]_q\) quantum code where
+  Using the code GRM\(_{q^2}(v,m)\) for \(0\leq v \leq q(m-1)-1\), the Hermitian construction yields a pure \([[q^{2m},q^{2m}-k(v),d(v^{\perp})]]_q\) quantum code where
   \(k(v) = \sum_{j=0}^{m}(-1)^{j}\dbinom{m}{j}\dbinom{m+v-jq^2}{v-jq^2}\) and \(d(v^{\perp}) = (R+1)q^{2Q}\) with \(v+1 = (q^2 - 1)Q + R\).
 
-  For a CSS code constructed from classical codes \(C_1\) and \(C_2\) the punctured code is defined as the classical code
+  For a CSS code constructed from classical codes \(C_1\) and \(C_2\), the punctured code is defined as the classical code
   \(P(C) = \{(a_ib_i)_{i=1}^{n}|a \in C_1, b \in C_2^{\perp}\}^{\perp}\). Quantum RM codes can be punctured to any length \(r\), provided
   \(P(C) = \mathcal{R}_q(v_2-v_1,m)\) has a codeword of this weight.
-#   A linear code \([n,k]_{q^{2}}\) contained in its Hermitian dual implies the existence of \([[n,n-2k,d]]_q\) quantum code where \(d = min(wt(C^{\perp_{h}}\C))\).
+#   A linear code \([n,k]_{q^{2}}\) contained in its Hermitian dual implies the existence of \([[n,n-2k,d]]_q\) quantum code where \(d = \min(wt(C^{\perp_{h}}\C))\).
 
 protection: |
-  QRM\(_{d}(m)\) quantum codes are \(\mathcal{M}_{d}^{m}\) distillation codes of distance \(D=2\). We define a \(\mathcal{M}_{d}^{m}\) distillation code as any \(n\) Galois-qudit stabilizer code \(C\) having the following properties: (a) All \(M \in \mathcal{M}_{d}^{m}\) are transversal so that \(M^{\otimes n}C(M^{\otimes n})^{\dag} = M_{L}^{\dag}CM_{L}\), (b) the code has distance \(D \geq 2\), and (c) the code has logical pauli operators \(X_{L} = X[\mathbf{1}]\) and \(Z_{L} = Z[(d-1)\mathbf{1}]\). \(\mathbf{1}\) is a shorthabd for the vector \((1,1, \hdots, 1)\).
+  QRM\(_{d}(m)\) quantum codes are \(\mathcal{M}_{d}^{m}\) distillation codes of distance \(D=2\). We define a \(\mathcal{M}_{d}^{m}\) distillation code as any \(n\) Galois-qudit stabilizer code \(C\) having the following properties: (a) All \(M \in \mathcal{M}_{d}^{m}\) are transversal so that \(M^{\otimes n}C(M^{\otimes n})^{\dagger} = M_{L}^{\dagger}CM_{L}\), (b) the code has distance \(D \geq 2\), and (c) the code has logical pauli operators \(X_{L} = X[\mathbf{1}]\) and \(Z_{L} = Z[(d-1)\mathbf{1}]\). Here, \(\mathbf{1}\) is a shorthand for the vector \((1,1, \dots, 1)\).
+
+# rate: '\(k(v) = \sum_{j=0}^{m}(-1)^{j}\dbinom{m}{j}\dbinom{m+v-jq}{v-jq}\) and \(n = q^{m}\) so rate must be \(\frac{\sum_{j=0}^{m}(-1)^{j}\dbinom{m}{j}\dbinom{m+v-jq}{v-jq}}{q^m}\).'
 
-features:
-  rate: '\(k(v) = \sum_{j=0}^{m}(-1)^{j}\dbinom{m}{j}\dbinom{m+v-jq}{v-jq}\) and \(n = q^{m}\) so rate must be \(\frac{\sum_{j=0}^{m}(-1)^{j}\dbinom{m}{j}\dbinom{m+v-jq}{v-jq}}{q^m}\).'
 
+features:
   transversal_gates: 'QRM\(_{d}(m)\) quantum codes are \(\mathcal{M}_{d}^{m}\) distillation codes of distance \(D=2\).'