~

codes/classical/q-ary_digits/ag/reed_solomon/rs/reed_solomon.yml

@@ -40,7 +40,7 @@ features:
     - 'Berlekamp-Welch decoder with runtime of \hyperref[topic:asymptotics]{order} \(O(n^3)\) \cite{manual:{E. R. Berlekamp and L. Welch, Error Correction of Algebraic Block Codes. U.S. Patent, Number 4,633,470 1986.}} (see exposition in Ref. \cite{doi:10.1016/0020-0190(92)90195-2}), assuming that \(t \geq (n+k)/2\).'
     - 'Gao decoder using extended Euclidean algorithm \cite{doi:10.1007/978-1-4757-3789-9_5}.'
     - 'Fast-Fourier-transform decoder with runtime of \hyperref[topic:asymptotics]{order} \(O(n \text{polylog}n)\) \cite{doi:10.1109/TIT.1978.1055816}.'
-    - 'List decoders try to find a low-degree bivariate polynomial \(Q(x,y)\) such that evaluation of \(Q\) at \((\alpha_i,y_i)\) is zero. By choosing proper degrees, it can be shown such polynomial exists by drawing an analogy between evaluation of \(Q(\alpha_i,y_i)\) and solving a homogenous linear equation (interpolation). Once this is done, one lists roots of \(y\) that agree at \(\geq t\) points. The breakthrough Sudan list-decoding algorithm corrects up to \(1-\sqrt{2R}\) fraction of errors asymptotically in \(n\) \cite{doi:10.1006/JCOM.1997.0439}. Roth and Ruckenstein proposed a modified key equation that allows for correction of more than \(\left\lfloor (n-k)/2 \right\rfloor\) errors \cite{doi:10.1109/18.817522}. The Guruswami-Sudan algorithm improved the Sudan algorithm to \(1-\sqrt{R}\) \cite{doi:10.1109/SFCS.1998.743426}, meaning that RS codes achieve list-decoding capacity; see Ref. \cite{doi:10.1109/TIT.2006.878164} for bounds. It was later shown that generic RS codes achieve list-decoding capacity \cite{arxiv:2206.05256}. A modification of the Guruswami-Sudan algorithm by Koetter and Vardy is used for soft-decision decoding \cite{doi:10.1109/TIT.2003.819332} (see also Ref. \cite{doi:10.1109/26.79287}). Subcodes of RS codes whose evaluation points lie in a \hyperref[topic:finite-fields]{subfield} can be decoded up to the \(1-R\) \cite{doi:10.1145/2488608.2488715}.'
+    - 'List decoders try to find a low-degree bivariate polynomial \(Q(x,y)\) such that evaluation of \(Q\) at \((\alpha_i,y_i)\) is zero. By choosing proper degrees, it can be shown such polynomial exists by drawing an analogy between evaluation of \(Q(\alpha_i,y_i)\) and solving a homogenous linear equation (interpolation). Once this is done, one lists roots of \(y\) that agree at \(\geq t\) points. The breakthrough Sudan list-decoding algorithm corrects up to \(1-\sqrt{2R}\) fraction of errors asymptotically in \(n\) \cite{doi:10.1006/JCOM.1997.0439}. Roth and Ruckenstein proposed a modified key equation that allows for correction of more than \(\left\lfloor (n-k)/2 \right\rfloor\) errors \cite{doi:10.1109/18.817522}. The Guruswami-Sudan algorithm improved the Sudan algorithm to \(1-\sqrt{R}\) \cite{doi:10.1109/SFCS.1998.743426}, meaning that RS codes achieve list-decoding capacity; see Ref. \cite{doi:10.1109/TIT.2006.878164} for bounds. It was later shown that generic RS codes achieve list-decoding capacity \cite{arxiv:2206.05256}. A modification of the Guruswami-Sudan algorithm by Koetter and Vardy is used for soft-decision decoding \cite{doi:10.1109/TIT.2003.819332} (see also Ref. \cite{doi:10.1109/26.79287}). Subcodes of RS codes whose evaluation points lie in a \hyperref[topic:finite-fields]{subfield} can be decoded up to the \(1-R\) \cite{doi:10.1145/2488608.2488715}. List decoding of RS codes is known as noisy polynomial interpolation in cryptography \cite{doi:10.1145/301250.301312}.'
     - 'The ubiquity of RS codes has yielded off-the-shelf VLSI intergrated-circuit decoding hardware \cite{doi:10.1109/92.953498} (see also Ref. \cite{doi:10.1109/9780470546345}, Ch. 5 and 10).'
 
 

codes/quantum/oscillators/oscillators.yml

@@ -22,7 +22,7 @@ description: |
 
   States can be represented by a series via a basis expansion, such as that in the countable basis of Fock states \(|n\rangle\) with \(n\geq 0\).
   Alternatively, states can be represented as functions over the reals by expanding in a continuous "basis" (more technically, set of tempered distributions in the space dual to Schwartz space), such as the position "basis" \(|y\rangle\) with \(y\in\mathbb{R}\) or the momentum "basis" \(|p\rangle\) with \(p\in\mathbb{R}\).
-  A third option is to use coherent states \(|\alpha\rangle\) with \(\alpha\in\mathbb{C}\), which are eigenstates of the annihilation operator, which correspond to classical electromagnetic signals, and which resolve the identity \cite{arxiv:math-ph/0210005,arxiv:10.1016/0034-4877(71)90006-1,doi:10.1103/PhysRevB.12.1118,doi:10.1103/PhysRevB.18.6744}.
+  A third option is to use coherent states \(|\alpha\rangle\) with \(\alpha\in\mathbb{C}\), which are eigenstates of the annihilation operator, which correspond to classical electromagnetic signals, and which resolve the identity \cite{arxiv:math-ph/0210005,doi:10.1016/0034-4877(71)90006-1,doi:10.1103/PhysRevB.12.1118,doi:10.1103/PhysRevB.18.6744}.
   States can further be represented as functions over the joint position-momentum phase space in the Wigner function formalism \cite{doi:10.1103/PhysRev.40.749,doi:10.1103/PhysRevA.15.449}.
 
   An important subset of states is formed by the \textit{Gaussian states}, which are in one-to-one correspondence with a (displacement) vector and covariance matrix \cite{arxiv:quant-ph/0410100,arxiv:0801.4604,arxiv:1110.3234,arxiv:2010.15518,arxiv:2409.11628}.