diff --git a/codes/classical/bits/covering/nearly_perfect.yml b/codes/classical/bits/covering/nearly_perfect.yml index 3c79f2f78..b66d30b6e 100644 --- a/codes/classical/bits/covering/nearly_perfect.yml +++ b/codes/classical/bits/covering/nearly_perfect.yml @@ -8,10 +8,10 @@ physical: bits logical: bits name: 'Nearly perfect code' -introduced: '\cite{doi:10.1016/0012-365X(72)90025-8,manual:{N. V. Semakov, V. A. Zinov''ev, G. V. Zaitsev, “Uniformly Packed Codes”, Probl. Peredachi Inf., 7:1 (1971), 38–50; Problems Inform. Transmission, 7:1 (1971), 30–39}}' +introduced: '\cite{doi:10.1109/TIT.1962.1057714,doi:10.1016/0012-365X(72)90025-8,manual:{N. V. Semakov, V. A. Zinov''ev, G. V. Zaitsev, “Uniformly Packed Codes”, Probl. Peredachi Inf., 7:1 (1971), 38–50; Problems Inform. Transmission, 7:1 (1971), 30–39}}' description: | - An \((n,K,2t+1)\) binary code is nearly perfect if parameters \(n\), \(K\), and \(t\) are such that the Johnson bound + An \((n,K,2t+1)\) binary code is nearly perfect if parameters \(n\), \(K\), and \(t\) are such that the Johnson bound \cite{doi:10.1109/TIT.1962.1057714}, \begin{align} \frac{{n \choose t}\left(\frac{n-t}{t+1}-\left\lfloor \frac{n-t}{t+1}\right\rfloor \right)}{\left\lfloor \frac{n}{t+1}\right\rfloor }+\sum_{j=0}^{t}{n \choose j}\leq2^{n}/K \end{align} diff --git a/codes/classical/q-ary_digits/q-ary_digits_into_q-ary_digits.yml b/codes/classical/q-ary_digits/q-ary_digits_into_q-ary_digits.yml index 76f511804..d7f6b7011 100644 --- a/codes/classical/q-ary_digits/q-ary_digits_into_q-ary_digits.yml +++ b/codes/classical/q-ary_digits/q-ary_digits_into_q-ary_digits.yml @@ -85,8 +85,8 @@ protection: | \end{defterm} \subsection{Bounds on code parameters} - Bounds on the parameters of an \((n,K,d)_q\) code include the Hamming a.k.a. sphere-packing bound, Singleton bound, Gilbert-Varshamov (GV) bound, Griesmer bound, Plotkin bound, and various linear programming (LP) bounds; see \cite{preset:HKSbasics}. - A code whose parameters attain the Hamming bound (Singleton bound, Griesmer bound, Delsarte LP bound) is called a perfect code (an MDS code, a Griesmer code, an \hyperref[code:univ_opt_q-ary]{LP universally optimal code}). + Bounds on the parameters of an \((n,K,d)_q\) code include the Hamming a.k.a. sphere-packing bound, Singleton bound, Gilbert-Varshamov (GV) bound, Griesmer bound, Plotkin bound, Johnson bound, and various linear programming (LP) bounds; see \cite{preset:HKSbasics}. + A code whose parameters attain the Hamming bound (Singleton bound, Griesmer bound, Johnson bound, Delsarte LP bound) is called a perfect code (an MDS code, a Griesmer code, a nearly perfect code, an \hyperref[code:univ_opt_q-ary]{LP universally optimal code}). \begin{defterm}{Gilbert-Varshamov (GV) bound} \label{topic:gv-bound} @@ -102,6 +102,7 @@ protection: | where \(h_q\) is the \(q\)-ary entropy function, \begin{align} h_{q}(\delta)=-\delta\log_{q}\frac{\delta}{q-1}-(1-\delta)\log_{q}(1-\delta)~. + \end{align} \end{defterm} features: diff --git a/codes/quantum/properties/block/quantum_mds.yml b/codes/quantum/properties/block/quantum_mds.yml index 3ac14edb9..cc6c8b696 100644 --- a/codes/quantum/properties/block/quantum_mds.yml +++ b/codes/quantum/properties/block/quantum_mds.yml @@ -12,9 +12,7 @@ short_name: 'Quantum MDS' introduced: '\cite{arxiv:quant-ph/9604034,arxiv:quant-ph/9702031,arxiv:quant-ph/9703048}' description: | - An \(((n,K,d))\) code constructed out of \(q\)-dimensional qubits or Galois qudits is an MDS code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the quantum Singleton bound is satisfied. - - In other words, the quantum Singleton bound \cite{arxiv:quant-ph/9604034,arxiv:quant-ph/9702031,arxiv:quant-ph/9703048} + An \(((n,K,d))\) code constructed out of \(q\)-dimensional qubits or Galois qudits is an MDS code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the quantum Singleton bound \cite{arxiv:quant-ph/9604034,arxiv:quant-ph/9702031,arxiv:quant-ph/9703048}, \begin{align} K \leq q^{n-2(d-1)} \end{align} diff --git a/codes/quantum/properties/block/quantum_perfect.yml b/codes/quantum/properties/block/quantum_perfect.yml index 24831538f..21dd31fdb 100644 --- a/codes/quantum/properties/block/quantum_perfect.yml +++ b/codes/quantum/properties/block/quantum_perfect.yml @@ -10,9 +10,7 @@ name: 'Perfect quantum code' #introduced: '' description: | - A \hyperref[topic:degeneracy]{non-degenerate} code constructed out of \(q\)-dimensional qudits and having parameters \(((n,K,2t+1))\) is perfect if \(n\), \(K\), \(t\), and \(q\) are such that the quantum Hamming bound is satisfied, - - In other words, the quantum Hamming bound \cite{arxiv:quant-ph/9602022} + A \hyperref[topic:degeneracy]{non-degenerate} code constructed out of \(q\)-dimensional qudits and having parameters \(((n,K,2t+1))\) is perfect if \(n\), \(K\), \(t\), and \(q\) are such that the quantum Hamming bound \cite{arxiv:quant-ph/9602022}, \begin{align} \sum_{j=0}^{t}(q^2-1)^{j}{n \choose j}\leq q^{n}/K \end{align}