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Copy file name to clipboardExpand all lines: codes/classical/analog/modulation/modulation.yml
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\textit{Quadrature amplitude modulation (QAM)} associates each point in a two-dimensional constellation with a complex-valued two-quadrature amplitude of a band-limited signal \cite[Ch. 16]{doi:10.1017/9781316822708}.
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notes:
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- 'See books \cite{manual:{J. G. Proakis and M. Salehi. Digital communications. Vol. 4. New York: McGraw-hill, 2001},doi:10.1016/B978-0-12-373580-5.X5033-9} for an introduction to modulation schemes.'
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- 'See books \cite{manual:{J. G. Proakis and M. Salehi, \emph{Digital communications}, vol. 4 (McGraw-Hill, 2001)},doi:10.1016/B978-0-12-373580-5.X5033-9} for an introduction to modulation schemes.'
Copy file name to clipboardExpand all lines: codes/classical/analog/sphere_packing/lattice/points_into_lattices.yml
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\end{align}
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The covering radius of a lattice is defined similarly as above, but with the spheres' \textit{covering radius} now being the smallest one such that the spheres cover all space.
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In general, finding the covering radius of a lattice is \(NP\)-hard \cite{manual:{van Emde, Boas P. "Another NP-complete partition problem and the complexity of computing short vectors in lattices." TR (1981)}}.
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In general, finding the covering radius of a lattice is \(NP\)-hard \cite{manual:{P. van Emde Boas, "Another NP-complete partition problem and the complexity of computing short vectors in lattices," Technical Report (1981)}}.
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The \textit{lattice quantizer problem} is to find a lattice whose \textit{fundamental Voronoi cell} \(\Pi\), the Voronoi cell at the origin, has the smallest possible normalized second moment,
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\begin{align}
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Higher-dimensional lattices yield quantizers with lower normalized second moments than the 1D integer lattice \cite{manual:{P. L. Zador, Development and evaluation of procedures for quantizing multivariate distributions, PhD thesis, Stanford Univ., 1963},doi:10.1109/TIT.1982.1056490}.
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The \textit{shortest vector problem} (SVP) asks for the shortest nonzero vector in a given lattice and is related to cryptographic protocols.
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Solving SVP up to an error independent of lattice dimension is NP-complete \cite{manual:{van Emde Boas, P. (1981). Another NP-complete problem and the complexity of computing short vectors in a lattice. Technical Report, Department of Mathematics, University of Amsterdam},doi:10.1137/S0097539700373039}.
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Solving SVP up to an error independent of lattice dimension is NP-complete \cite{manual:{P. van Emde Boas, "Another NP-complete problem and the complexity of computing short vectors in a lattice," Technical Report, Department of Mathematics, University of Amsterdam (1981)},doi:10.1137/S0097539700373039}.
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The Lenstra-Lenstra-Lovasz (LLL) algorithm solves SVP in polynomial time, but up to an error exponential in the dimension \cite{doi:10.1007/BF01457454}; see the book \cite{doi:10.1007/978-0-387-77993-5}.
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# Lattices are also characterized by the \textit{minimal norm} \(\mu\), which is the minimal norm of a nonzero vector in the lattice.
- 'See books \cite{doi:10.1007/978-1-4757-6568-7,doi:10.1007/b98975,manual:{C. A. Rogers. Packing and covering. Vol. 54. Cambridge: University Press, 1964}} for introductions and overviews of lattices.'
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- 'See books \cite{doi:10.1007/978-1-4757-6568-7,doi:10.1007/b98975,manual:{C. A. Rogers, \emph{Packing and covering}, vol. 54 (Cambridge University Press, 1964)}} for introductions and overviews of lattices.'
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- 'See LMFDB \cite{preset:LMFDB} and Catalogue of Lattices \cite{manual:{G. Nebe and N. J. A. Sloane. "Catalogue of Lattices." \url{https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/index.html}}} for databases of lattices.'
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- 'Tables of bounds on kissing numbers \cite{manual:{H. Cohn. "Kissing numbers." \url{https://cohn.mit.edu/kissing-numbers}}}. Popular summary of bounds on kissing numbers in 17-21 dimensions in \href{https://www.quantamagazine.org/mathematicians-discover-new-way-for-spheres-to-kiss-20250115/}{Quanta Magazine}.'
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- 'See Refs. \cite{manual:{J. Cannon, W. Bosma, C. Fieker, and A. Steel (2008). HANDBOOK OF MAGMA FUNCTIONS},doi:10.1145/190347.190362,doi:10.1006/jsco.1996.0125} for various examples and implementations in Magma.'
Copy file name to clipboardExpand all lines: codes/classical/analog/sphere_packing/lattice/root/dn/dthree.yml
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The \(D_3\) fcc lattice exhibits the densest packing and highest kissing number of 12 in three dimensions.
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The \textit{Kepler conjecture} \cite{manual:{J. Kepler. The six-cornered snowflake. Paul Dry Books, 2010}} states that the \(D_3\) fcc lattice has the densest packing in 3D.
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The \textit{Kepler conjecture} \cite{manual:{J. Kepler, \emph{The six-cornered snowflake} (Paul Dry Books, 2010)}} states that the \(D_3\) fcc lattice has the densest packing in 3D.
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It was first proven that this lattice was the densest 3D lattice packing \cite{manual:{C. F. Gauss, Besprechung des Buchs von L. A. Seeber, Untersuchungen uber die Eigenschaften der positiven terndren quadratischen Formen usw. Gottingsche Gelehrte Anzeigen, July 9, 1831 = Werke, II, pp. 18g8-196, 1876}}.
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Determining the maximum density of any sphere packing in 3D was then reduced to a computationally tractable problem \cite{doi:10.1007/978-3-642-65234-9}, which was solved \cite{arxiv:math/9811071} and formalized in automated proof checking software \cite{arxiv:1501.02155}.
Member of the family of \([2m+2,m+1]\) double circulant codes such that \(m\) is prime of the form \(8k+3\) for some \(k\), and \(2m+2\) is a multiple of eight.
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See \cite[Ch. 16]{preset:MacSlo} for their generator matrix.
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Karlin codes can be mapped to extended cyclic and extended quadratic-residue codes over \(\mathbb{F}_4\) \cite[Ch. 16]{preset:MacSlo}\cite{doi:10.1006/ffta.2001.0333,manual:{Karlin M., MacWilliams F. J. Quadratic residue codes over GF(4) and their binary images. InIEEE Int. Symp. on Information Theory, Asilomar, CA 1972}}\cite[Sec. 2.4.2]{doi:10.1007/3-540-30731-1} by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) \cite{doi:10.1006/ffta.2001.0333}.
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Karlin codes can be mapped to extended cyclic and extended quadratic-residue codes over \(\mathbb{F}_4\) \cite[Ch. 16]{preset:MacSlo}\cite{doi:10.1006/ffta.2001.0333,manual:{M. Karlin and F. J. MacWilliams, "Quadratic residue codes over GF(4) and their binary images," in IEEE Int. Symp. on Information Theory, Asilomar, CA, 1972}}\cite[Sec. 2.4.2]{doi:10.1007/3-540-30731-1} by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) \cite{doi:10.1006/ffta.2001.0333}.
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relations:
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parents:
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- code_id: binary_linear
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- code_id: quasi_cyclic
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detail: 'Karlin codes can be mapped to extended cyclic and extended quadratic-residue codes over \(\mathbb{F}_4\) \cite[Ch. 16]{preset:MacSlo}\cite{doi:10.1006/ffta.2001.0333,manual:{Karlin M., MacWilliams F. J. Quadratic residue codes over GF(4) and their binary images. InIEEE Int. Symp. on Information Theory, Asilomar, CA 1972}}\cite[Sec. 2.4.2]{doi:10.1007/3-540-30731-1} by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) \cite{doi:10.1006/ffta.2001.0333}.'
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detail: 'Karlin codes can be mapped to extended cyclic and extended quadratic-residue codes over \(\mathbb{F}_4\) \cite[Ch. 16]{preset:MacSlo}\cite{doi:10.1006/ffta.2001.0333,manual:{M. Karlin and F. J. MacWilliams, "Quadratic residue codes over GF(4) and their binary images," in IEEE Int. Symp. on Information Theory, Asilomar, CA, 1972}}\cite[Sec. 2.4.2]{doi:10.1007/3-540-30731-1} by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) \cite{doi:10.1006/ffta.2001.0333}.'
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- code_id: self_dual
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detail: 'Karlin codes are Euclidean self-dual doubly even codes \cite[Ch. 16]{preset:MacSlo}, and some of them are extremal \cite{doi:10.1016/S0019-9958(78)90158-4,doi:10.1109/18.945245}.'
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cousins:
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- code_id: q-ary_quad_residue
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detail: 'Karlin codes can be mapped to extended cyclic and extended quadratic-residue codes over \(\mathbb{F}_4\) \cite[Ch. 16]{preset:MacSlo}\cite{doi:10.1006/ffta.2001.0333,manual:{Karlin M., MacWilliams F. J. Quadratic residue codes over GF(4) and their binary images. InIEEE Int. Symp. on Information Theory, Asilomar, CA 1972}}\cite[Sec. 2.4.2]{doi:10.1007/3-540-30731-1} by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) \cite{doi:10.1006/ffta.2001.0333}.'
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detail: 'Karlin codes can be mapped to extended cyclic and extended quadratic-residue codes over \(\mathbb{F}_4\) \cite[Ch. 16]{preset:MacSlo}\cite{doi:10.1006/ffta.2001.0333,manual:{M. Karlin and F. J. MacWilliams, "Quadratic residue codes over GF(4) and their binary images," in IEEE Int. Symp. on Information Theory, Asilomar, CA, 1972}}\cite[Sec. 2.4.2]{doi:10.1007/3-540-30731-1} by identifying \((0,\omega,\bar{\omega},1)\) with \((00),(10),(01),(11)\) \cite{doi:10.1006/ffta.2001.0333}.'
Copy file name to clipboardExpand all lines: codes/classical/bits/cyclic/quad_residue/binary_quad_residue.yml
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features:
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decoders:
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- 'Algebraic decoder \cite{manual:{Y. H. Chen, T. K. Truong, Y. Chang, C. D. Lee, and S. H. Chen (2007). "Algebraic decoding of quadratic residue codes using Berlekamp-Massey algorithm". Journal of information science and engineering, 23(1), 127-145}}.'
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- 'Algebraic decoder \cite{manual:{Y. H. Chen, T. K. Truong, Y. Chang, C. D. Lee, and S. H. Chen, "Algebraic decoding of quadratic residue codes using Berlekamp-Massey algorithm," Journal of information science and engineering 23(1), 127-145 (2007)}}.'
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- 'Introduction of quadratic-residue codes in Refs. \cite{preset:MacSlo,doi:10.1017/CBO9780511807077}\cite[Sec. 2.7]{preset:HKScyclic}.'
Copy file name to clipboardExpand all lines: codes/classical/bits/easy/checksum/crc.yml
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- 'CRC-16 and CRC-32 are used in data transmission, e.g., IEEE 802.16e, IEEE 802.3 \cite{manual:{C. Borrelli, "IEEE 802.3 cyclic redundancy check." application note: Virtex Series and Virtex-II Family, XAPP209 (v1. 0) (2001)}} and TCP/IP communication \cite[Sec. 2.3.3]{doi:10.1007/978-0-387-68192-4_2}.'
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- 'See Ref. \cite{doi:10.1109/CICN.2015.108} and book \cite{manual:{R. B. Wells. Applied coding and information theory for engineers. Prentice-Hall, Inc., 1998}} for introductions to CRC codes.'
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- 'See Ref. \cite{doi:10.1109/CICN.2015.108} and book \cite{manual:{R. B. Wells, \emph{Applied coding and information theory for engineers} (Prentice-Hall, 1998)}} for introductions to CRC codes.'
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- 'See Refs. \cite{doi:10.1109/DSN.2002.1028931,doi:10.1109/DSN.2004.1311885} for exhaustive lists of CRC polynomials, as well as the \href{https://users.ece.cmu.edu/~koopman/crc/crc32.html}{CRC Polynomial Zoo website} by Philip Koopman.'
Copy file name to clipboardExpand all lines: codes/classical/bits/easy/unary.yml
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- 'Neural networks \cite{arxiv:1009.4495}.'
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- 'Birdsong production \cite{manual:{I. R. Fiete and H. S. Seung (2007). "Neural network models of birdsong production, learning, and coding". New Encyclopedia of Neuroscience. Eds. L. Squire, T. Albright, F. Bloom, F. Gage, and N. Spitzer. Elsevier}}.'
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- 'Birdsong production \cite{manual:{I. R. Fiete and H. S. Seung, "Neural network models of birdsong production, learning, and coding," in \emph{New Encyclopedia of Neuroscience}, eds. L. Squire, T. Albright, F. Bloom, F. Gage, and N. Spitzer (Elsevier, 2007)}}.'
Copy file name to clipboardExpand all lines: codes/classical/bits/fountain/luby_transform.yml
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features:
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- 'Sum-Product Algorithm (SPA), often called a peeling decoder \cite{doi:10.1017/CBO9780511791338,manual:{D. J. C. MacKay. 2002. Information Theory, Inference & Learning Algorithms. Cambridge University Press, USA}}, similar to belief propagation \cite{doi:10.1145/3501714.3501727}.'
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- 'Sum-Product Algorithm (SPA), often called a peeling decoder \cite{doi:10.1017/CBO9780511791338,manual:{D. J. C. MacKay, \emph{Information Theory, Inference and Learning Algorithms} (Cambridge University Press, 2002)}}, similar to belief propagation \cite{doi:10.1145/3501714.3501727}.'
Copy file name to clipboardExpand all lines: codes/classical/bits/nonlinear/gray_map/duals/preparata.yml
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The union of a shortened Preparata code and some of its translates forms a Hamming code \cite[pg. 475]{preset:MacSlo}.
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- code_id: extended_hamming
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detail: 'Any code with the same parameters as the Preparata code must be a distance invariant subcode of a (possibly nonlinear) code with the same parameters as the extended Hamming code \cite{preset:Semakov71,manual:{G. V. Zaitsev, V. A. Zinoviev, and N. V. Semakov (1973). "Interrelation of Preparata and Hamming codes and extension of Hamming codes to new double-error-correcting codes". In Proc. 2nd International Symp. Inform. Theory (pp. 257-263)}}.'
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detail: 'Any code with the same parameters as the Preparata code must be a distance invariant subcode of a (possibly nonlinear) code with the same parameters as the extended Hamming code \cite{preset:Semakov71,manual:{G. V. Zaitsev, V. A. Zinoviev, and N. V. Semakov, "Interrelation of Preparata and Hamming codes and extension of Hamming codes to new double-error-correcting codes," in Proc. 2nd International Symp. Inform. Theory (1973): 257–263}}.'
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- code_id: combinatorial_design
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detail: 'Preparata codewords of each weight form 3-designs, and the minimum-weight codewords yield infinite families of 4-designs, including Steiner 4-designs with block sizes 5 and 6 \cite[Rem. 5.5.6 and Thms. 5.5.7, 5.5.11]{preset:HKSdesigns}\cite[pg. 471]{preset:MacSlo}.'
Copy file name to clipboardExpand all lines: codes/classical/bits/nonlinear/gray_map/originals/nordstrom_robinson.yml
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detail: 'The NR code is an orthogonal array of strength \(5\) \cite[pg. 141]{preset:MacSlo}.'
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- code_id: octacode
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detail: 'The NR code is the image of the octacode under the \term{Gray map} \cite[Sec. 6.3]{preset:HKSrings}\cite{manual:{G. D. Forney Jr., N. J. A. Sloane, and M. D. Trott. "The Nordstrom-Robinson code is the binary image of the octacode." In Coding and Quantization: DIMACS/IEEE workshop 1992 Oct 19 (pp. 19-26). American Mathematical Society},doi:10.1142/3603}\cite[Thm. 12]{arxiv:math/0207208}. The \((14, 64, 6)\) shortened NR code is the image of the heptacode under the \term{Gray map} \cite[Exam. 5]{doi:10.1109/TIT.2021.3114636}.'
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detail: 'The NR code is the image of the octacode under the \term{Gray map} \cite[Sec. 6.3]{preset:HKSrings}\cite{manual:{G. D. Forney Jr., N. J. A. Sloane, and M. D. Trott, "The Nordstrom-Robinson code is the binary image of the octacode," in Coding and Quantization: DIMACS/IEEE workshop 1992 Oct 19 (pp. 19–26). American Mathematical Society},doi:10.1142/3603}\cite[Thm. 12]{arxiv:math/0207208}. The \((14, 64, 6)\) shortened NR code is the image of the heptacode under the \term{Gray map} \cite[Exam. 5]{doi:10.1109/TIT.2021.3114636}.'
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- code_id: biorthogonal
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detail: 'The NR code is the union of eight cosets of a linear \([16,5,8]\) code, i.e., the first-order Reed-Muller (biorthogonal) code \cite[pgs. 76 and 476]{preset:MacSlo}.'
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- code_id: extended_golay
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detail: 'The NR code can be constructed using the extended Golay code by first selecting a set of codewords satisfying certain conditions and then deleting specific coordinates \cite[pg. 73]{preset:MacSlo}.'
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- code_id: self_dual
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The NR code is self-dual in that its distance distribution is invariant under the \hyperref[topic:weight-enumerator]{MacWilliams transform} \cite{doi:10.1109/TIT.1983.1056676}.
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It maps to the octacode, a self-dual code over \(\mathbb{Z}_4\) under the \term{Gray map} \cite[Sec. 6.3]{preset:HKSrings}\cite{arxiv:math/9310227,manual:{G. D. Forney Jr., N. J. A. Sloane, and M. D. Trott. "The Nordstrom-Robinson code is the binary image of the octacode." In Coding and Quantization: DIMACS/IEEE workshop 1992 Oct 19 (pp. 19-26). American Mathematical Society},doi:10.1142/3603}.
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It maps to the octacode, a self-dual code over \(\mathbb{Z}_4\) under the \term{Gray map} \cite[Sec. 6.3]{preset:HKSrings}\cite{arxiv:math/9310227,manual:{G. D. Forney Jr., N. J. A. Sloane, and M. D. Trott, "The Nordstrom-Robinson code is the binary image of the octacode," in Coding and Quantization: DIMACS/IEEE workshop 1992 Oct 19 (pp. 19–26). American Mathematical Society},doi:10.1142/3603}.
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- code_id: nearly_perfect
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detail: 'The punctured \((15,256,5)\) NR code saturates the Johnson bound and is therefore nearly perfect \cite[Exam. 5.5.5]{preset:HKSdesigns}.'
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