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2 changes: 1 addition & 1 deletion codes/classical/analog/analog.yml
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- 'Each signal point is assigned its own Voronoi cell, and a received point is mapped back to the center of the Voronoi cell in which it is located upon reception.'

notes:
- 'Database of sphere packings \cite{manual:{Agrell, E. "Database of sphere packings, 2019." Online: http://codes. se/packings. Accessed (2015).}}.'
- 'Database of sphere packings \cite{manual:{Agrell, E. "Database of sphere packings, 2019." Online: https://codes. se/packings. Accessed (2015).}}.'
- 'See Refs. \cite{doi:10.1109/18.720549,arxiv:cs/0611112} for reviews of sphere packing.'


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2 changes: 1 addition & 1 deletion codes/classical/analog/lattice/points_into_lattices.yml
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notes:
- 'See books \cite{doi:10.1007/978-1-4757-6568-7,doi:10.1007/b98975} for introductions and overviews of lattices.'
- 'See LMFDB \cite{manual:{The LMFDB Collaboration, The L-functions and modular forms database, https://www.lmfdb.org, 2024.}} and Catalogue of Lattices \cite{manual:{G. Nebe and N. J. A. Sloane. "Catalogue of Lattices." http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/index.html}} for databases of lattices.'
- 'See LMFDB \cite{manual:{The LMFDB Collaboration, The L-functions and modular forms database, https://www.lmfdb.org, 2024.}} and Catalogue of Lattices \cite{manual:{G. Nebe and N. J. A. Sloane. "Catalogue of Lattices." https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/index.html}} for databases of lattices.'
- 'Tables of bounds on kissing numbers \cite{manual:{H. Cohn. "Kissing numbers." https://cohn.mit.edu/kissing-numbers}}.'
- 'See Refs. \cite{manual:{Cannon, J., Bosma, W., Fieker, C., & Steel, A. (2008). HANDBOOK OF MAGMA FUNCTIONS.},doi:10.1145/190347.190362,doi:10.1006/jsco.1996.0125} for various examples and implementations in Magma.'

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2 changes: 1 addition & 1 deletion codes/classical/analog/modulation/ppm.yml
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An analog code encoding into \(q\) different signals such that each codeword corresponds to a signal.
realizations:
- 'Greek hydraulic semaphore system \cite{manual:{Michael Lahanas. "Ancient Greek Communication Methods". https://web.archive.org/web/20141102224501/http://www.mlahanas.de/Greeks/Communication.htm. Archived from the original on 2014-11-02.},doi:10.1049/iet-com.2019.0051}.'
- 'Greek hydraulic semaphore system \cite{manual:{Michael Lahanas. "Ancient Greek Communication Methods". https://web.archive.org/web/20141102224501/https://www.mlahanas.de/Greeks/Communication.htm. Archived from the original on 2014-11-02.},doi:10.1049/iet-com.2019.0051}.'
- 'Telegraph time-division multiplexing.'
- 'Radio-control, fiber-optic communications, and deep-space communications.'

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2 changes: 1 addition & 1 deletion codes/classical/bits/binary_linear.yml
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# Decoding being NP hard is discussed in Bruss Leuchs

notes:
- 'Tables of bounds and examples of linear codes for various \(n\) and \(k\), extending code tables by A. E. Brouwer \cite{manual:{Andries E. Brouwer, Bounds on linear codes, in: Vera S. Pless and W. Cary Huffman (Eds.), Handbook of Coding Theory, pp. 295-461, Elsevier, 1998.}}, are maintained by M. Grassl at this \href{http://codetables.markus-grassl.de/}{website}.'
- 'Tables of bounds and examples of linear codes for various \(n\) and \(k\), extending code tables by A. E. Brouwer \cite{manual:{Andries E. Brouwer, Bounds on linear codes, in: Vera S. Pless and W. Cary Huffman (Eds.), Handbook of Coding Theory, pp. 295-461, Elsevier, 1998.}}, are maintained by M. Grassl at this \href{https://codetables.markus-grassl.de/}{website}.'

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2 changes: 1 addition & 1 deletion codes/classical/bits/cyclic/bch.yml
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notes:
- 'See books \cite{preset:MacSlo,preset:LinCostello,doi:10.1017/CBO9780511807077} for expositions on BCH codes and code tables.'
- 'See Kaiserslautern database \cite{preset:KLdatabase} for explicit codes.'
- 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Cyclic Codes (BCH-Bound).” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CCyclic-BCHBound.html}}.'
- 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Cyclic Codes (BCH-Bound).” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. https://mint.sbg.ac.at/desc_CCyclic-BCHBound.html}}.'

realizations:
- 'Satellite communication \cite{manual:{Cheung, K-M., and F. Pollara. "Phobos lander coding system: Software and analysis." The Telecommunications and Data Acquisition Report (1988).}}'
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2 changes: 1 addition & 1 deletion codes/classical/bits/easy/dual_hamming/simplex.yml
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detail: '\(q\)-ary simplex codes reduce to simplex codes for \(q=2\).'
cousins:
- code_id: extended_reed_solomon
detail: 'Simplex codes are extended RS codes \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Simplex Code.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CSimplex.html}}.'
detail: 'Simplex codes are extended RS codes \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Simplex Code.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. https://mint.sbg.ac.at/desc_CSimplex.html}}.'
- code_id: hamming
detail: 'Hamming and simplex codes are dual to each other.'
- code_id: dual
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2 changes: 1 addition & 1 deletion codes/classical/bits/fountain/raptor.yml
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- 'Raptor codes can be decoded using inactivation decoding \cite{arxiv:1706.05814}, a combination of belief-propogation and Gaussian elimination decoding.'

realizations:
- 'Two versions of Raptor codes have been standardized by IETF: \href{https://datatracker.ietf.org/doc/html/rfc5053}{R10} and the more recent \href{http://tools.ietf.org/html/rfc6330}{RaptorQ}. RaptorQ is used in mobile multimedia broadcasts as specified in ETSI technical specifications. It is also used in the mobile \href{https://www.atsc.org/wp-content/uploads/2016/01/A331S33-174r6-Signaling-Delivery-Sync-FEC.pdf}{Next Gen TV} standard.'
- 'Two versions of Raptor codes have been standardized by IETF: \href{https://datatracker.ietf.org/doc/html/rfc5053}{R10} and the more recent \href{https://tools.ietf.org/html/rfc6330}{RaptorQ}. RaptorQ is used in mobile multimedia broadcasts as specified in ETSI technical specifications. It is also used in the mobile \href{https://www.atsc.org/wp-content/uploads/2016/01/A331S33-174r6-Signaling-Delivery-Sync-FEC.pdf}{Next Gen TV} standard.'
- 'Raptor codes are useful in scenarios where erasure (i.e. weak signal or noisy channel) is common, such as in military or disaster scenarios.'

notes:
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of Lee weight 3/ 4 \cite{doi:10.1109/18.312154,arxiv:math/0207208}.'

notes:
- 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Preparata Codes.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CPreparata.html}}.'
- 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Preparata Codes.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. https://mint.sbg.ac.at/desc_CPreparata.html}}.'


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notes:
- 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Kerdock Codes.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CKerdock.html}}.'
- 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Kerdock Codes.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. https://mint.sbg.ac.at/desc_CKerdock.html}}.'


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# It is an orthogonal array of strength 5.

# cite following
# Orthogonal array specification -- http://mint.sbg.ac.at/desc_CNordstromRobinson.html#bib-CNordstromRobinson-3
# Orthogonal array specification -- https://mint.sbg.ac.at/desc_CNordstromRobinson.html#bib-CNordstromRobinson-3

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- '''Find erasures and Decode'' a.k.a. Viderman''s algorithm correcting \hyperref[topic:asymptotics]{order} \(\Omega(n)\) errors in \hyperref[topic:asymptotics]{order} \(O(n)\) time \cite{doi:10.1145/2493252.2493255}.'

fault_tolerance:
- 'The flip decoding algorithm is fault tolerant against parity check errors \cite{doi:10.1109/18.556668}; see also \href{http://courses.csail.mit.edu/6.440/spring08/index.html}{course notes} by M. Sudan.'
- 'The flip decoding algorithm is fault tolerant against parity check errors \cite{doi:10.1109/18.556668}; see also \href{https://courses.csail.mit.edu/6.440/spring08/index.html}{course notes} by M. Sudan.'

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2 changes: 1 addition & 1 deletion codes/classical/matrices/raid/array.yml
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- code_id: distributed_storage
cousins:
- code_id: reed_solomon
detail: 'RS codes over \(q=2^m\) are used in RAID 6 \cite{manual:{Anvin, H. Peter. "The mathematics of RAID-6." (2007).},manual:{S. T. Position. (2009) Common raid disk data format specification. [Online]. Available: http://www.snia.org/tech activities/standards/curr standards/ddf}}; see \cite{preset:HPArray}.'
detail: 'RS codes over \(q=2^m\) are used in RAID 6 \cite{manual:{Anvin, H. Peter. "The mathematics of RAID-6." (2007).},manual:{S. T. Position. (2009) Common raid disk data format specification. [Online]. Available: https://www.snia.org/tech activities/standards/curr standards/ddf}}; see \cite{preset:HPArray}.'


# Begin Entry Meta Information
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description: |
A block code of length \(n\) is quasi-cyclic if, for each codeword \(c_1 \cdots c_{\ell} c_{\ell+1} \cdots c_n\), the string \(c_{n-\ell+1} \cdots c_n c_1 \cdots c_{n-\ell}\), where each entry is cyclically shifted by \(\ell\) increments, is also a codeword.
The generator of an \([mn_0,mk_0]\) quasi-cyclic linear code is representable as a block matrix of \(m \times m\) circulant matrices \cite{manual:{Thomas A. Gulliver, \href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.143.3623}{Construction of quasi-cyclic codes}, Thesis, University of New Brunswick, 1989.}}.
The generator of an \([mn_0,mk_0]\) quasi-cyclic linear code is representable as a block matrix of \(m \times m\) circulant matrices \cite{manual:{Thomas A. Gulliver, \href{https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.143.3623}{Construction of quasi-cyclic codes}, Thesis, University of New Brunswick, 1989.}}.
Quasi-cyclic codes can also be understood in terms of the number of automorphism-group orbits required to generate all codewords.
All codewords of a cyclic code can be obtained from any codeword via cyclic shifts, meaning that the code consists of only one orbit.
On the other hand, quasi-cyclic codes consist of multiple disjoint orbits, meaning that not all of their codewords can be obtained from each other.
notes:
- 'A database of quasi-cyclic codes with searchable parameters such as block length and dimension is constructed and displayed \href{http://www.tec.hkr.se/~chen/research/codes/qc.htm}{here}.'
- 'A database of quasi-cyclic codes with searchable parameters such as block length and dimension is constructed and displayed \href{https://www.tec.hkr.se/~chen/research/codes/qc.htm}{here}.'

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2 changes: 1 addition & 1 deletion codes/classical/q-ary_digits/ag/evaluation_varieties.yml
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#TO DO: is projective same as evaluation?
# https://link.springer.com/article/10.1007/s10623-021-00905-4
# https://algo.epfl.ch/_media/en/courses/2008-2009/mct_l04.pdf
# http://people.sabanciuniv.edu/~mlavrauw/LaStVa2009b.pdf
# https://people.sabanciuniv.edu/~mlavrauw/LaStVa2009b.pdf
# Basic Notions: Thm. 1.1.6


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Their automorphism groups have been identified \cite{doi:10.1016/0097-3165(87)90060-4}.
notes:
- 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Extended Reed–Solomon Code.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CReedSolomon-extended.html}}.'
- 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Extended Reed–Solomon Code.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. https://mint.sbg.ac.at/desc_CReedSolomon-extended.html}}.'

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- 'RS codes as outer codes concatenated with convolutional codes are used indirectly in space exploration programs such as Voyager and Galileo. RS codes were part of a telemetry channel coding standard issued by the Consultative Committee for Space Data Systems (see Ref. \cite{doi:10.1109/9780470546345}, Ch. 3).'
- 'Automatic repeat request (ARQ) data transmission protocols (see Ref. \cite{doi:10.1109/9780470546345}, Ch. 7).'
- 'Slow-frequency-hop spread-spectrum transmission (see Ref. \cite{doi:10.1109/9780470546345}, Chs. 8-9).'
- 'RS codes over \(q=2^m\) are used in RAID 6 \cite{manual:{Anvin, H. Peter. "The mathematics of RAID-6." (2007).},manual:{S. T. Position. (2009) Common raid disk data format specification. [Online]. Available: http://www.snia.org/tech activities/standards/curr standards/ddf}}; see \cite{preset:HPArray}.'
- 'RS codes over \(q=2^m\) are used in RAID 6 \cite{manual:{Anvin, H. Peter. "The mathematics of RAID-6." (2007).},manual:{S. T. Position. (2009) Common raid disk data format specification. [Online]. Available: https://www.snia.org/tech activities/standards/curr standards/ddf}}; see \cite{preset:HPArray}.'
- 'Coded sharding designs in blockchains to increase efficiency \cite{arxiv:1809.10361}.'
- 'Used in QR-Codes to retrieve damaged barcodes \cite{manual:{International Organization for Standardization, Information Technology: Automatic Identification and Data Capture Techniques-QR Code 2005 Bar Code Symbology Specification, 2nd ed., IEC18004 (ISO, 2006).}}.'
- 'Wireless communication systems such as 3G, DVB, and WiMAX \cite{doi:10.1109/PIMRC.2010.5672091}.'
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notes:
- 'See Kaiserslautern database \cite{preset:KLdatabase} for explicit codes.'
- 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Reed–Solomon Code.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CReedSolomon.html}}.'
- 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Reed–Solomon Code.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. https://mint.sbg.ac.at/desc_CReedSolomon.html}}.'
- 'Popular summary in \href{https://www.quantamagazine.org/how-mathematical-curves-power-cryptography-20220919/}{Quanta Magazine}.'


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- code_id: tamo_barg
detail: 'Tamo-Barg codes reduce to RS codes when \(r=k\).'
- code_id: reed_solomon_nrt
detail: 'RS NRT codes reduce to RS codes when the NRT metric is equivalent to the Hamming metric \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Reed-Solomon Codes for OOAs.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_OReedSolomon.html}}.'
detail: 'RS NRT codes reduce to RS codes when the NRT metric is equivalent to the Hamming metric \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Reed-Solomon Codes for OOAs.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. https://mint.sbg.ac.at/desc_OReedSolomon.html}}.'
cousins:
- code_id: dna
detail: 'RS codes have been used for DNA storage \cite{doi:10.1002/anie.201411378}.'
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2 changes: 1 addition & 1 deletion codes/classical/q-ary_digits/alternant/bch/q-ary_bch.yml
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notes:
- 'See books \cite{preset:MacSlo,preset:LinCostello,doi:10.1017/CBO9780511807077} for expositions on BCH codes and code tables.'
- 'See Kaiserslautern database \cite{preset:KLdatabase} for explicit codes.'
- 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Cyclic Codes (BCH-Bound).” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CCyclic-BCHBound.html}}.'
- 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Cyclic Codes (BCH-Bound).” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. https://mint.sbg.ac.at/desc_CCyclic-BCHBound.html}}.'

realizations:
- 'DVDs, disk drives, and two-dimensional bar codes \cite{doi:10.1109/TIT.2019.2913389}.'
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An NRT analogue of an RS code.
notes:
- 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Reed-Solomon Codes for OOAs.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. http://mint.sbg.ac.at/desc_OReedSolomon.html}}.'
- 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Reed-Solomon Codes for OOAs.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2024-09-05. https://mint.sbg.ac.at/desc_OReedSolomon.html}}.'


relations:
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2 changes: 1 addition & 1 deletion codes/classical/q-ary_digits/dual/self_dual.yml
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notes:
- See books \cite{doi:10.1007/3-540-30731-1,doi:10.1017/CBO9780511807077} for more on self-dual codes.
- 'See Refs. \cite{doi:10.1016/j.disc.2004.06.010,arxiv:1706.01694} for constructions of binary self-dual codes.'
- 'See \href{http://www.unilim.fr/pages_perso/philippe.gaborit/SD/index.html}{Tables of Self-Dual Codes} for a database of self-dual codes over \(GF(2)\), \(GF(3)\), \(GF(4)\) (Euclidean or Hermitian), \(GF(5)\), and \(GF(7)\). See also Ref. \cite{doi:10.1016/S1071-5797(03)00011-X}.'
- 'See \href{https://www.unilim.fr/pages_perso/philippe.gaborit/SD/index.html}{Tables of Self-Dual Codes} for a database of self-dual codes over \(GF(2)\), \(GF(3)\), \(GF(4)\) (Euclidean or Hermitian), \(GF(5)\), and \(GF(7)\). See also Ref. \cite{doi:10.1016/S1071-5797(03)00011-X}.'

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2 changes: 1 addition & 1 deletion codes/classical/q-ary_digits/easy/hexacode.yml
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notes:
- 'See \cite[Sec. 10.3]{doi:10.1017/CBO9780511807077} for an exposition.'
- 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Hexacode.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. http://mint.sbg.ac.at/desc_CHexa.html}}.'
- 'See corresponding MinT database entry \cite{manual:{Rudolf Schürer and Wolfgang Ch. Schmid. “Hexacode.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2015-09-03. https://mint.sbg.ac.at/desc_CHexa.html}}.'

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