From 89c2c93bff335279e24ca2e3a14eb28259b7077e Mon Sep 17 00:00:00 2001 From: VVA2024 Date: Sat, 13 Jul 2024 17:51:41 -0400 Subject: [PATCH] ~ --- codes/classical/bits/nonlinear/constantin_rao.yml | 6 +++--- codes/classical/bits/nonlinear/vt_single_deletion.yml | 6 ++---- .../properties/block/universally_optimal/t-designs.yml | 2 +- 3 files changed, 6 insertions(+), 8 deletions(-) diff --git a/codes/classical/bits/nonlinear/constantin_rao.yml b/codes/classical/bits/nonlinear/constantin_rao.yml index 7db0eed02..04adc147f 100644 --- a/codes/classical/bits/nonlinear/constantin_rao.yml +++ b/codes/classical/bits/nonlinear/constantin_rao.yml @@ -14,7 +14,7 @@ introduced: '\cite{doi:10.1016/S0019-9958(79)90329-2}' description: |- A nonlinear single-asymmetric-error code that generalize VT codes and that is constructed from an Abelian group. - A CR code for group \(G\) and fixed group element \(g\) consists of all binary strings \(c=c_1c_2\cdotsc_n\) that satisfy \(\sum_i^n c_i g_i = g\) for some elements \(g_i\) \cite[Def. 1.3]{arxiv:1310.7536}. + A CR code for group \(G\) and fixed group element \(g\) consists of all binary strings \(c=c_1c_2\cdots c_n\) that satisfy \(\sum_i c_i g_i = g\) for some elements \(g_i\) \cite[Def. 1.3]{arxiv:1310.7536}. Here, addition is the group operation, the multiplication \(1 g_i = g_i\), and \(0 g_i = 0_G\) where \(0_G\) is the identity element. CR codes can be generalized to the \(q\)-ary case and also to codes correcting more than one asymmetric error \cite{manual:{Kløve, Torleiv. Error correcting codes for the asymmetric channel. Department of Pure Mathematics, University of Bergen, 1981.}}. @@ -25,8 +25,8 @@ protection: | features: rate: | - CR codes for particular groups have higher rates than single-error-correcting codes under the binary asymmetric channel for all lengths except \(n = 2^r - 1\), in which case CR codes reduce to Hamming codes \cite{manual:{Kløve, Torleiv. Error correcting codes for the asymmetric channel. Department of Pure Mathematics, University of Bergen, 1981.}}; see Ref. \cite{arxiv:1310.7536}. - Size analysis is presented in Refs. \cite{doi:10.1016/S0019-9958(80)90082-0,doi:10.1109/18.651063}.' + CR codes for particular groups have higher rates than distance-one codes under the binary asymmetric channel for all lengths except \(n = 2^r - 1\), in which case CR codes reduce to Hamming codes \cite{manual:{Kløve, Torleiv. Error correcting codes for the asymmetric channel. Department of Pure Mathematics, University of Bergen, 1981.}}; see Ref. \cite{arxiv:1310.7536}. + Size analysis is presented in Refs. \cite{doi:10.1016/S0019-9958(80)90082-0,doi:10.1109/18.651063}. relations: diff --git a/codes/classical/bits/nonlinear/vt_single_deletion.yml b/codes/classical/bits/nonlinear/vt_single_deletion.yml index 95a0c6be9..478caf687 100644 --- a/codes/classical/bits/nonlinear/vt_single_deletion.yml +++ b/codes/classical/bits/nonlinear/vt_single_deletion.yml @@ -19,6 +19,7 @@ description: |- C_{n,a}=\left\{x\in\{0,1\}^n: \sum_{i=1}^n i~x_i = a\mod (n+1) \right\}. \end{align} + By adapting a construction of Tenengolts \cite{manual:{G. M. Tenengolts, \emph{Class of codes correcting bit loss and errors in the preceding bit} (translated to English), Automation and Remote Control, 37(5), 797–802 (1976).}}, VT codes can be modified to yield slightly longer linear codes \cite{arxiv:math/0207197}. VT codes can be generalized to the \(q\)-ary case \cite{arxiv:1708.04071,doi:10.1109/TIT.1984.1056962,arxiv:1906.07887}. protection: 'Corrects a single asymmetric error (a \(0\) mapped to a \(1\)), a single deletion, or a single insertion of an arbitrary bit in an arbitrary position for any choice of \(a\).' @@ -34,10 +35,7 @@ relations: - code_id: constantin_rao detail: 'CR codes for \(G=\mathbb{Z}_{n+1}\) reduce to VT codes.' - code_id: insertion_deletion - cousins: - - code_id: binary_linear - detail: 'By adapting a construction of Tenengolts \cite{manual:{G. M. Tenengolts, \emph{Class of codes correcting bit loss and errors in the preceding bit} (translated to English), Automation and Remote Control, 37(5), 797–802 (1976).}}, VT codes can be modified to yield slightly longer linear codes \cite{arxiv:math/0207197}.' - + # Begin Entry Meta Information _meta: diff --git a/codes/classical/properties/block/universally_optimal/t-designs.yml b/codes/classical/properties/block/universally_optimal/t-designs.yml index 417e36212..98fa22a3e 100644 --- a/codes/classical/properties/block/universally_optimal/t-designs.yml +++ b/codes/classical/properties/block/universally_optimal/t-designs.yml @@ -20,7 +20,7 @@ description: | \begin{align} \int_{X}dxp(x)={\textstyle \frac{1}{|D|}}\sum_{x\in D}p(x)~, \end{align} - where the integral is over \(X\) (given some measure \(d x\)), while the sum is over the design \(D\). + where the integral is over \(X\) (given some measure \(d x\)), while the sum is over the design \(D\subset X\). A \textit{weighted design} is a design for which each term \(p(x)\) in the above sum must be multiplied by a weight \(w(x)\) in order to be equal to the left-hand side. The most well-known examples of designs are (exact) quadrature/cubature formulas for integration over the reals \cite{manual:{Stroud, Arthur H. Approximate calculation of multiple integrals. Prentice Hall, 1971.},doi:10.1017/S0962492900002701,doi:10.1016/S0885-064X(03)00011-6,doi:10.18434/M3167}, \(X = \mathbb{R}^D\) (with appropriate measure); these tend to be weighted designs.