From be4fc5c7239374b322face8981f9d1003c63f864 Mon Sep 17 00:00:00 2001 From: "Victor V. Albert" Date: Thu, 23 Nov 2023 08:09:58 -0500 Subject: [PATCH] gray cousins --- codes/classical/bits/nonlinear/gray.yml | 5 +++-- codes/classical/bits/nonlinear/kerdock.yml | 2 ++ codes/classical/bits/nonlinear/preparata.yml | 2 ++ codes/classical/bits/nonlinear/sphere_packing/julin12.yml | 2 ++ codes/classical/bits/reed_muller.yml | 2 ++ codes/classical/rings/pentacode.yml | 2 ++ 6 files changed, 13 insertions(+), 2 deletions(-) diff --git a/codes/classical/bits/nonlinear/gray.yml b/codes/classical/bits/nonlinear/gray.yml index aa9a27021..992c9345b 100644 --- a/codes/classical/bits/nonlinear/gray.yml +++ b/codes/classical/bits/nonlinear/gray.yml @@ -16,8 +16,9 @@ description: | A simple example is the case \(n=2\), also known as the \textit{Gray map}, which produces the ordering \(0\to 00\), \(1\to 01\), \(2\to 11\), and \(3\to 10\). The Gray map differs in the last two numbers from the usual binary expansion of the natural numbers, which maps \(0\to 00\), \(1\to 01\), \(2\to 10\), and \(3\to 11\). - Layout out the Gray-map output strings counterclockwise on the corners of a 1D square, gray codes have been generalized such that nearest-neighbor strings differ by only one digit when the strings are arranged in higher-dimensional hypercubes \cite{doi:10.1002/j.1538-7305.1958.tb03887.x}. - Further generalizations called \textit{combinatorial Gray codes} \cite{doi:10.1137/0209013} refer to methods to generate organize combinatorial objects such that successive objects differ in some particular way. Particular \(q\)-ary extensions \cite{doi:10.1109/TSMCB.2012.2210706} of Gray codes may be useful in digital imaging and signal processing. + Gray codes have been generalized such that nearest-neighbor strings differ by only one digit when the strings are arranged in higher-dimensional hypercubes \cite{doi:10.1002/j.1538-7305.1958.tb03887.x}. + Further generalizations called \textit{combinatorial Gray codes} \cite{doi:10.1137/0209013} refer to methods to organize combinatorial objects such that successive objects differ in some particular way. + Particular \(q\)-ary extensions \cite{doi:10.1109/TSMCB.2012.2210706} of Gray codes may be useful in digital imaging and signal processing. features: encoders: diff --git a/codes/classical/bits/nonlinear/kerdock.yml b/codes/classical/bits/nonlinear/kerdock.yml index 9c2d505e9..250109dfe 100644 --- a/codes/classical/bits/nonlinear/kerdock.yml +++ b/codes/classical/bits/nonlinear/kerdock.yml @@ -41,6 +41,8 @@ relations: cousins: - 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: gray + 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 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\((1,m)\).' diff --git a/codes/classical/bits/nonlinear/preparata.yml b/codes/classical/bits/nonlinear/preparata.yml index adbc70052..0f3efd30f 100644 --- a/codes/classical/bits/nonlinear/preparata.yml +++ b/codes/classical/bits/nonlinear/preparata.yml @@ -40,6 +40,8 @@ relations: 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: gray + 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: quasi_perfect detail: 'Punctured Preparata codes are quasi-perfect \cite[pg. 475]{preset:MacSlo}.' - code_id: reed_muller diff --git a/codes/classical/bits/nonlinear/sphere_packing/julin12.yml b/codes/classical/bits/nonlinear/sphere_packing/julin12.yml index 581a65f34..5c98c2743 100644 --- a/codes/classical/bits/nonlinear/sphere_packing/julin12.yml +++ b/codes/classical/bits/nonlinear/sphere_packing/julin12.yml @@ -36,6 +36,8 @@ relations: detail: 'Using Construction \(A\), the Julin-Golay codes yield non-lattice sphere-packings that hold records in 9 and 11 dimensions.' - code_id: q-ary_over_zq detail: 'Julin codes can be obtained from simple nonlinear codes over \(\mathbb{Z}_4\) using the Gray map \cite{doi:10.1007/BF01388558}.' + - code_id: gray + detail: 'Julin codes can be obtained from simple nonlinear codes over \(\mathbb{Z}_4\) using the Gray map \cite{doi:10.1007/BF01388558}.' # Begin Entry Meta Information diff --git a/codes/classical/bits/reed_muller.yml b/codes/classical/bits/reed_muller.yml index a3bba9fe2..b18ad8e75 100644 --- a/codes/classical/bits/reed_muller.yml +++ b/codes/classical/bits/reed_muller.yml @@ -57,6 +57,8 @@ relations: detail: 'RM\(^*(r,m)\) codes are equivalent to subcodes of BCH codes of designed distance \(2^{m-r}-1\) while RM\((r,m)\) are subcodes of extended BCH codes of the same designed distance \cite[Ch. 13]{preset:MacSlo}.' - code_id: quaternary_over_z4 detail: 'RM codes are images of linear quaternary codes over \(\mathbb{Z}_4\) under the Gray map \cite[Sec. 6.3]{preset:HKSrings}.' + - code_id: gray + detail: 'RM codes are images of linear quaternary codes over \(\mathbb{Z}_4\) under the Gray map \cite[Sec. 6.3]{preset:HKSrings}.' - code_id: dual detail: 'The codes RM\((r,m)\) and RM\((m-r-1,m)\) are dual to each other.' - code_id: binary_duadic diff --git a/codes/classical/rings/pentacode.yml b/codes/classical/rings/pentacode.yml index e6f22044b..bd67fca94 100644 --- a/codes/classical/rings/pentacode.yml +++ b/codes/classical/rings/pentacode.yml @@ -21,6 +21,8 @@ relations: cousins: - code_id: best detail: 'Codewords of the Best code can be obtained by applying the Gray map to the pentacode \cite[Sec. 2]{doi:10.1007/BF01388558}.' + - code_id: gray + detail: 'Codewords of the Best code can be obtained by applying the Gray map to the pentacode \cite[Sec. 2]{doi:10.1007/BF01388558}.' # Begin Entry Meta Information