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curl -X POST \
  'http://laksa014.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://ncatlab.org/nlab/show/error+correcting+code\')), getAhrefsUnparsedNoserviceFromURL(\'https://ncatlab.org/nlab/show/error+correcting+code\'))) FORMAT JSONEachRow'

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{"found_url":"https:\/\/ncatlab.org\/nlab\/show\/error+correcting+code","crawl_time":1775905917,"first_indexed_time":1620318312,"http_code":200,"src_unparsed":"org,ncatlab!\/nlab\/show\/error+correcting+code s443","src_root_hash":"4727794731343683214","history_drop_reason":null,"meta_title":"error correcting code in nLab","meta_descriptions":[],"attrs_boilerpipe_text":"Contents\nContents\nIdea\nExamples\nReferences\nGeneral\nRelation to 2d CFT\nIdea\nIn\ncoding theory\n, an\nerror correcting code\nis a means to encode data in a way that is robust against errors (\nnoise\n).\nVery broadly, for\nL\nL\na\nfinite set\nplaying the role of a\nspace of states\nthat is to be saved\/communicated\/analyzed, an error correcting code for\nL\nL\nis an\ninjection\nL\n↪\nP\nL \\overset{\\;\\;\\;}{\\hookrightarrow} P\ninto a larger set. The idea is that noise\/errors move the image of\nL\nL\nwithin\nP\nP\n, but if\nP\nP\nis large enough and the embedding chosen well enough, then a sufficiently small number of errors stays within a small neighbourhood of\nL\nL\nin\nP\nP\nthat allows to\nretract\nback to\nL\nL\n.\nThe simplest example is the\nrepetition code\n, where the inclusion is the\ndiagonal\non the\nn\nn\n-fold\nCartesian product\nL\n↪\ndiag\nP\n≔\nL\n×\n⋯\n×\nL\n⏟\nn\nfactors\nℓ\n↦\n(\nℓ\n,\n⋯\n,\nℓ\n)\n.\n\\array{\n  L \n  &\n    \\overset{diag}{\\hookrightarrow}\n  & \n  P \\coloneqq \\underset{n \\; factors}{\\underbrace{L \\times \\cdots \\times L}}\n  \\\\\n  \\ell &\\mapsto& (\\ell, \\cdots, \\ell)\n  }\n  \\,.\nThis code “protects against\nn\n\/\n2\n−\n1\nn\/2-1\nerrors” in an evident sense.\nMuch attention in\ncoding theory\nis instead on the special class of\nlinear codes\n, where\nL\nL\nand\nP\nP\ncarry the structure of\nvector spaces\n(necessarily over a\nfinite field\nif they are\nfinite sets\nof relevance in practice) and where the inclusion\nL\n↪\nP\nL \\hookrightarrow P\nis a\nlinear map\n.\nExamples\nlinear code\nHamming code\nbinary linear code\nbinary Golay code\nquantum error correcting code\nbit flip code\nstabilizer code\nsurface code\nHaPPY code\nMajorana dimer code\ntoric code\nReferences\nGeneral\nSee also the references at\ncoding theory\nand\nlinear code\n.\nVictor V. Albert\net al.,\nerrorcorrectionzoo.org\nN. J. A. Sloane\n,\nError-Correcting Codes and Cryptography\n, The Mathematical Gardner, D. A. Klarner (editor), Prindle, Weber & Schmidt, Boston, MA, 1981, pp. 346-382, Reprinted in “yptologia’’, Vol. 6 (1982), 128-153 and 258-278.\nAn observation on classical codes preconceiving aspects of\nholographic tensor network\nquantum error correcting codes\n:\nBeni Yoshida\n,\nInformation storage capacity of discrete spin systems\n, Annals of Physics 338, 134 (2013) (\narXiv:1111.3275\n)\nRelation to 2d CFT\nConstruction of chiral\n2d SCFTs\nfrom error-correcting codes:\nDavide Gaiotto\n,\nTheo Johnson-Freyd\n,\nHolomorphic SCFTs with small index\n, Canadian Journal of Mathematics ,\n74\n2 (2022) 573-601 �\narXiv:1811.00589\n,\ndoi:10.4153\/S0008414X2100002X\n�\nOn their\nelliptic genera\nKohki Kawabata, Shinichiro Yahagi,\nElliptic genera from classical error-correcting codes\n�\narXiv:2308.12592\n�\nLast revised on February  3, 2026 at 11:46:41.\n    See the\nhistory\nof this page for a list of all contributions to it.","attrs_markdown":"# nLab error correcting code\n[Skip the Navigation Links](https:\/\/ncatlab.org\/nlab\/show\/error+correcting+code#navEnd) \\|   [Home Page](https:\/\/ncatlab.org\/nlab\/show\/HomePage \"Home page\") \\| [All Pages](https:\/\/ncatlab.org\/nlab\/all_pages \"List of all pages\") \\| [Latest Revisions](https:\/\/ncatlab.org\/nlab\/latest_revisions \"Latest edits and page creations\") \\| [Discuss this page](https:\/\/nforum.ncatlab.org\/discussion\/12802\/#Item_4 \"Discuss this page in its dedicated thread on the nForum\") \\|\n\n# Contents\n- [Idea](https:\/\/ncatlab.org\/nlab\/show\/error+correcting+code#idea)\n- [Examples](https:\/\/ncatlab.org\/nlab\/show\/error+correcting+code#examples)\n- [References](https:\/\/ncatlab.org\/nlab\/show\/error+correcting+code#references)\n  - [General](https:\/\/ncatlab.org\/nlab\/show\/error+correcting+code#general)\n  - [Relation to 2d CFT](https:\/\/ncatlab.org\/nlab\/show\/error+correcting+code#relation_to_2d_cft)\n## Idea\nIn [coding theory](https:\/\/ncatlab.org\/nlab\/show\/coding+theory), an *error correcting code* is a means to encode data in a way that is robust against errors ([noise](https:\/\/ncatlab.org\/nlab\/show\/noise)).\n\nVery broadly, for L L a [finite set](https:\/\/ncatlab.org\/nlab\/show\/finite+set) playing the role of a [space of states](https:\/\/ncatlab.org\/nlab\/show\/space+of+states) that is to be saved\/communicated\/analyzed, an error correcting code for L L is an [injection](https:\/\/ncatlab.org\/nlab\/show\/injection) L ↪ P L \\\\overset{\\\\;\\\\;\\\\;}{\\\\hookrightarrow} P into a larger set. The idea is that noise\/errors move the image of L L within P P, but if P P is large enough and the embedding chosen well enough, then a sufficiently small number of errors stays within a small neighbourhood of L L in P P that allows to [retract](https:\/\/ncatlab.org\/nlab\/show\/retraction) back to L L.\n\nThe simplest example is the *repetition code*, where the inclusion is the [diagonal](https:\/\/ncatlab.org\/nlab\/show\/diagonal) on the n n\\-fold [Cartesian product](https:\/\/ncatlab.org\/nlab\/show\/Cartesian+product)\n\nL\n\n↪\n\ndiag\n\nP\n\n≔\n\nL\n\n×\n\n⋯\n\n×\n\nL\n\n⏟\n\nn\n\nfactors\n\nℓ\n\n↦\n\n(\n\nℓ\n\n,\n\n⋯\n\n,\n\nℓ\n\n)\n\n.\n\n\\\\array{ L & \\\\overset{diag}{\\\\hookrightarrow} & P \\\\coloneqq \\\\underset{n \\\\; factors}{\\\\underbrace{L \\\\times \\\\cdots \\\\times L}} \\\\\\\\ \\\\ell &\\\\mapsto& (\\\\ell, \\\\cdots, \\\\ell) } \\\\,.\n\nThis code “protects against n \/ 2 − 1 n\/2-1 errors” in an evident sense.\n\nMuch attention in [coding theory](https:\/\/ncatlab.org\/nlab\/show\/coding+theory) is instead on the special class of *[linear codes](https:\/\/ncatlab.org\/nlab\/show\/linear+codes)*, where L L and P P carry the structure of [vector spaces](https:\/\/ncatlab.org\/nlab\/show\/vector+spaces) (necessarily over a [finite field](https:\/\/ncatlab.org\/nlab\/show\/finite+field) if they are [finite sets](https:\/\/ncatlab.org\/nlab\/show\/finite+sets) of relevance in practice) and where the inclusion L ↪ P L \\\\hookrightarrow P is a [linear map](https:\/\/ncatlab.org\/nlab\/show\/linear+map).\n\n## Examples\n- [linear code](https:\/\/ncatlab.org\/nlab\/show\/linear+code)\n  - [Hamming code](https:\/\/ncatlab.org\/nlab\/show\/Hamming+code)\n  - [binary linear code](https:\/\/ncatlab.org\/nlab\/show\/binary+linear+code)\n    - [binary Golay code](https:\/\/ncatlab.org\/nlab\/show\/binary+Golay+code)\n- [quantum error correcting code](https:\/\/ncatlab.org\/nlab\/show\/quantum+error+correcting+code)\n  - [bit flip code](https:\/\/ncatlab.org\/nlab\/show\/bit+flip+code)\n  - [stabilizer code](https:\/\/ncatlab.org\/nlab\/show\/stabilizer+code)\n  - [surface code](https:\/\/ncatlab.org\/nlab\/show\/surface+code)\n  - [HaPPY code](https:\/\/ncatlab.org\/nlab\/show\/HaPPY+code)\n  - [Majorana dimer code](https:\/\/ncatlab.org\/nlab\/show\/Majorana+dimer+code)\n  - [toric code](https:\/\/ncatlab.org\/nlab\/show\/toric+code)\n## References\n### General\nSee also the references at *[coding theory](https:\/\/ncatlab.org\/nlab\/show\/coding+theory)* and *[linear code](https:\/\/ncatlab.org\/nlab\/show\/linear+code)*.\n\n- [Victor V. Albert](https:\/\/ncatlab.org\/nlab\/show\/Victor+V.+Albert) et al., *[errorcorrectionzoo.org](https:\/\/errorcorrectionzoo.org\/)*\n- [N. J. A. Sloane](https:\/\/ncatlab.org\/nlab\/show\/N.+J.+A.+Sloane), *Error-Correcting Codes and Cryptography*, The Mathematical Gardner, D. A. Klarner (editor), Prindle, Weber & Schmidt, Boston, MA, 1981, pp. 346-382, Reprinted in “yptologia’’, Vol. 6 (1982), 128-153 and 258-278.\n\nAn observation on classical codes preconceiving aspects of [holographic tensor network](https:\/\/ncatlab.org\/nlab\/show\/holographic+tensor+network) [quantum error correcting codes](https:\/\/ncatlab.org\/nlab\/show\/quantum+error+correcting+codes):\n\n- [Beni Yoshida](https:\/\/ncatlab.org\/nlab\/show\/Beni+Yoshida), *Information storage capacity of discrete spin systems*, Annals of Physics 338, 134 (2013) ([arXiv:1111.3275](https:\/\/arxiv.org\/abs\/1111.3275))\n### Relation to 2d CFT\nConstruction of chiral [2d SCFTs](https:\/\/ncatlab.org\/nlab\/show\/2d+SCFTs) from error-correcting codes:\n\n- [Davide Gaiotto](https:\/\/ncatlab.org\/nlab\/show\/Davide+Gaiotto), [Theo Johnson-Freyd](https:\/\/ncatlab.org\/nlab\/show\/Theo+Johnson-Freyd), *Holomorphic SCFTs with small index*, Canadian Journal of Mathematics , **74** 2 (2022) 573-601 �[arXiv:1811.00589](https:\/\/arxiv.org\/abs\/1811.00589), [doi:10.4153\/S0008414X2100002X](https:\/\/doi.org\/10.4153\/S0008414X2100002X)�\n\nOn their [elliptic genera](https:\/\/ncatlab.org\/nlab\/show\/elliptic+genera)\n\n- Kohki Kawabata, Shinichiro Yahagi, *Elliptic genera from classical error-correcting codes* �[arXiv:2308.12592](https:\/\/arxiv.org\/abs\/2308.12592)�\n\nLast revised on February 3, 2026 at 11:46:41. See the [history](https:\/\/ncatlab.org\/nlab\/history\/error+correcting+code) of this page for a list of all contributions to it.\n\n[Edit](https:\/\/ncatlab.org\/nlab\/edit\/error+correcting+code)[Discuss](https:\/\/nforum.ncatlab.org\/discussion\/12802\/#Item_4)[Previous revision](https:\/\/ncatlab.org\/nlab\/revision\/error+correcting+code\/12)[Changes from previous revision](https:\/\/ncatlab.org\/nlab\/show\/diff\/error+correcting+code)[History (12 revisions)](https:\/\/ncatlab.org\/nlab\/history\/error+correcting+code) [Cite](https:\/\/ncatlab.org\/nlab\/show\/error+correcting+code\/cite) [Print](https:\/\/ncatlab.org\/nlab\/print\/error+correcting+code) [Source](https:\/\/ncatlab.org\/nlab\/source\/error+correcting+code)","attrs_readable_markdown":"## Contents\n- [Idea](https:\/\/ncatlab.org\/nlab\/show\/error+correcting+code#idea)\n- [Examples](https:\/\/ncatlab.org\/nlab\/show\/error+correcting+code#examples)\n- [References](https:\/\/ncatlab.org\/nlab\/show\/error+correcting+code#references)\n  - [General](https:\/\/ncatlab.org\/nlab\/show\/error+correcting+code#general)\n  - [Relation to 2d CFT](https:\/\/ncatlab.org\/nlab\/show\/error+correcting+code#relation_to_2d_cft)\n## Idea\nIn [coding theory](https:\/\/ncatlab.org\/nlab\/show\/coding+theory), an *error correcting code* is a means to encode data in a way that is robust against errors ([noise](https:\/\/ncatlab.org\/nlab\/show\/noise)).\n\nVery broadly, for L L a [finite set](https:\/\/ncatlab.org\/nlab\/show\/finite+set) playing the role of a [space of states](https:\/\/ncatlab.org\/nlab\/show\/space+of+states) that is to be saved\/communicated\/analyzed, an error correcting code for L L is an [injection](https:\/\/ncatlab.org\/nlab\/show\/injection) L ↪ P L \\\\overset{\\\\;\\\\;\\\\;}{\\\\hookrightarrow} P into a larger set. The idea is that noise\/errors move the image of L L within P P, but if P P is large enough and the embedding chosen well enough, then a sufficiently small number of errors stays within a small neighbourhood of L L in P P that allows to [retract](https:\/\/ncatlab.org\/nlab\/show\/retraction) back to L L.\n\nThe simplest example is the *repetition code*, where the inclusion is the [diagonal](https:\/\/ncatlab.org\/nlab\/show\/diagonal) on the n n\\-fold [Cartesian product](https:\/\/ncatlab.org\/nlab\/show\/Cartesian+product)\n\nL ↪ diag P ≔ L × ⋯ × L ⏟ n factors ℓ ↦ ( ℓ , ⋯ , ℓ ) . \\\\array{ L & \\\\overset{diag}{\\\\hookrightarrow} & P \\\\coloneqq \\\\underset{n \\\\; factors}{\\\\underbrace{L \\\\times \\\\cdots \\\\times L}} \\\\\\\\ \\\\ell &\\\\mapsto& (\\\\ell, \\\\cdots, \\\\ell) } \\\\,.\n\nThis code “protects against n \/ 2 − 1 n\/2-1 errors” in an evident sense.\n\nMuch attention in [coding theory](https:\/\/ncatlab.org\/nlab\/show\/coding+theory) is instead on the special class of *[linear codes](https:\/\/ncatlab.org\/nlab\/show\/linear+codes)*, where L L and P P carry the structure of [vector spaces](https:\/\/ncatlab.org\/nlab\/show\/vector+spaces) (necessarily over a [finite field](https:\/\/ncatlab.org\/nlab\/show\/finite+field) if they are [finite sets](https:\/\/ncatlab.org\/nlab\/show\/finite+sets) of relevance in practice) and where the inclusion L ↪ P L \\\\hookrightarrow P is a [linear map](https:\/\/ncatlab.org\/nlab\/show\/linear+map).\n\n## Examples\n- [linear code](https:\/\/ncatlab.org\/nlab\/show\/linear+code)\n  - [Hamming code](https:\/\/ncatlab.org\/nlab\/show\/Hamming+code)\n  - [binary linear code](https:\/\/ncatlab.org\/nlab\/show\/binary+linear+code)\n    - [binary Golay code](https:\/\/ncatlab.org\/nlab\/show\/binary+Golay+code)\n- [quantum error correcting code](https:\/\/ncatlab.org\/nlab\/show\/quantum+error+correcting+code)\n  - [bit flip code](https:\/\/ncatlab.org\/nlab\/show\/bit+flip+code)\n  - [stabilizer code](https:\/\/ncatlab.org\/nlab\/show\/stabilizer+code)\n  - [surface code](https:\/\/ncatlab.org\/nlab\/show\/surface+code)\n  - [HaPPY code](https:\/\/ncatlab.org\/nlab\/show\/HaPPY+code)\n  - [Majorana dimer code](https:\/\/ncatlab.org\/nlab\/show\/Majorana+dimer+code)\n  - [toric code](https:\/\/ncatlab.org\/nlab\/show\/toric+code)\n## References\n### General\nSee also the references at *[coding theory](https:\/\/ncatlab.org\/nlab\/show\/coding+theory)* and *[linear code](https:\/\/ncatlab.org\/nlab\/show\/linear+code)*.\n\n- [Victor V. Albert](https:\/\/ncatlab.org\/nlab\/show\/Victor+V.+Albert) et al., *[errorcorrectionzoo.org](https:\/\/errorcorrectionzoo.org\/)*\n- [N. J. A. Sloane](https:\/\/ncatlab.org\/nlab\/show\/N.+J.+A.+Sloane), *Error-Correcting Codes and Cryptography*, The Mathematical Gardner, D. A. Klarner (editor), Prindle, Weber & Schmidt, Boston, MA, 1981, pp. 346-382, Reprinted in “yptologia’’, Vol. 6 (1982), 128-153 and 258-278.\n\nAn observation on classical codes preconceiving aspects of [holographic tensor network](https:\/\/ncatlab.org\/nlab\/show\/holographic+tensor+network) [quantum error correcting codes](https:\/\/ncatlab.org\/nlab\/show\/quantum+error+correcting+codes):\n\n- [Beni Yoshida](https:\/\/ncatlab.org\/nlab\/show\/Beni+Yoshida), *Information storage capacity of discrete spin systems*, Annals of Physics 338, 134 (2013) ([arXiv:1111.3275](https:\/\/arxiv.org\/abs\/1111.3275))\n### Relation to 2d CFT\nConstruction of chiral [2d SCFTs](https:\/\/ncatlab.org\/nlab\/show\/2d+SCFTs) from error-correcting codes:\n\n- [Davide Gaiotto](https:\/\/ncatlab.org\/nlab\/show\/Davide+Gaiotto), [Theo Johnson-Freyd](https:\/\/ncatlab.org\/nlab\/show\/Theo+Johnson-Freyd), *Holomorphic SCFTs with small index*, Canadian Journal of Mathematics , **74** 2 (2022) 573-601 �[arXiv:1811.00589](https:\/\/arxiv.org\/abs\/1811.00589), [doi:10.4153\/S0008414X2100002X](https:\/\/doi.org\/10.4153\/S0008414X2100002X)�\n\nOn their [elliptic genera](https:\/\/ncatlab.org\/nlab\/show\/elliptic+genera)\n\n- Kohki Kawabata, Shinichiro Yahagi, *Elliptic genera from classical error-correcting codes* �[arXiv:2308.12592](https:\/\/arxiv.org\/abs\/2308.12592)�\n\nLast revised on February 3, 2026 at 11:46:41. See the [history](https:\/\/ncatlab.org\/nlab\/history\/error+correcting+code) of this page for a list of all contributions to it.","meta_canonical":null}

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URL	https://ncatlab.org/nlab/show/error+correcting+code
Last Crawled	2026-04-11 11:11:57 (5 days ago)
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Boilerpipe Text	Contents Contents Idea Examples References General Relation to 2d CFT Idea In coding theory , an error correcting code is a means to encode data in a way that is robust against errors ( noise ). Very broadly, for L L a finite set playing the role of a space of states that is to be saved/communicated/analyzed, an error correcting code for L L is an injection L ↪ P L \overset{\;\;\;}{\hookrightarrow} P into a larger set. The idea is that noise/errors move the image of L L within P P , but if P P is large enough and the embedding chosen well enough, then a sufficiently small number of errors stays within a small neighbourhood of L L in P P that allows to retract back to L L . The simplest example is the repetition code , where the inclusion is the diagonal on the n n -fold Cartesian product L ↪ diag P ≔ L × ⋯ × L ⏟ n factors ℓ ↦ ( ℓ , ⋯ , ℓ ) . \array{ L & \overset{diag}{\hookrightarrow} & P \coloneqq \underset{n \; factors}{\underbrace{L \times \cdots \times L}} \\ \ell &\mapsto& (\ell, \cdots, \ell) } \,. This code “protects against n / 2 − 1 n/2-1 errors” in an evident sense. Much attention in coding theory is instead on the special class of linear codes , where L L and P P carry the structure of vector spaces (necessarily over a finite field if they are finite sets of relevance in practice) and where the inclusion L ↪ P L \hookrightarrow P is a linear map . Examples linear code Hamming code binary linear code binary Golay code quantum error correcting code bit flip code stabilizer code surface code HaPPY code Majorana dimer code toric code References General See also the references at coding theory and linear code . Victor V. Albert et al., errorcorrectionzoo.org N. J. A. Sloane , Error-Correcting Codes and Cryptography , The Mathematical Gardner, D. A. Klarner (editor), Prindle, Weber & Schmidt, Boston, MA, 1981, pp. 346-382, Reprinted in “yptologia’’, Vol. 6 (1982), 128-153 and 258-278. An observation on classical codes preconceiving aspects of holographic tensor network quantum error correcting codes : Beni Yoshida , Information storage capacity of discrete spin systems , Annals of Physics 338, 134 (2013) ( arXiv:1111.3275 ) Relation to 2d CFT Construction of chiral 2d SCFTs from error-correcting codes: Davide Gaiotto , Theo Johnson-Freyd , Holomorphic SCFTs with small index , Canadian Journal of Mathematics , 74 2 (2022) 573-601 � arXiv:1811.00589 , doi:10.4153/S0008414X2100002X � On their elliptic genera Kohki Kawabata, Shinichiro Yahagi, Elliptic genera from classical error-correcting codes � arXiv:2308.12592 � Last revised on February 3, 2026 at 11:46:41. See the history of this page for a list of all contributions to it.
Markdown	# nLab error correcting code [Skip the Navigation Links](https://ncatlab.org/nlab/show/error+correcting+code#navEnd) \\| [Home Page](https://ncatlab.org/nlab/show/HomePage "Home page") \\| [All Pages](https://ncatlab.org/nlab/all_pages "List of all pages") \\| [Latest Revisions](https://ncatlab.org/nlab/latest_revisions "Latest edits and page creations") \\| [Discuss this page](https://nforum.ncatlab.org/discussion/12802/#Item_4 "Discuss this page in its dedicated thread on the nForum") \\| # Contents - [Idea](https://ncatlab.org/nlab/show/error+correcting+code#idea) - [Examples](https://ncatlab.org/nlab/show/error+correcting+code#examples) - [References](https://ncatlab.org/nlab/show/error+correcting+code#references) - [General](https://ncatlab.org/nlab/show/error+correcting+code#general) - [Relation to 2d CFT](https://ncatlab.org/nlab/show/error+correcting+code#relation_to_2d_cft) ## Idea In [coding theory](https://ncatlab.org/nlab/show/coding+theory), an error correcting code is a means to encode data in a way that is robust against errors ([noise](https://ncatlab.org/nlab/show/noise)). Very broadly, for L L a [finite set](https://ncatlab.org/nlab/show/finite+set) playing the role of a [space of states](https://ncatlab.org/nlab/show/space+of+states) that is to be saved/communicated/analyzed, an error correcting code for L L is an [injection](https://ncatlab.org/nlab/show/injection) L ↪ P L \\overset{\\;\\;\\;}{\\hookrightarrow} P into a larger set. The idea is that noise/errors move the image of L L within P P, but if P P is large enough and the embedding chosen well enough, then a sufficiently small number of errors stays within a small neighbourhood of L L in P P that allows to [retract](https://ncatlab.org/nlab/show/retraction) back to L L. The simplest example is the repetition code, where the inclusion is the [diagonal](https://ncatlab.org/nlab/show/diagonal) on the n n\-fold [Cartesian product](https://ncatlab.org/nlab/show/Cartesian+product) L ↪ diag P ≔ L × ⋯ × L ⏟ n factors ℓ ↦ ( ℓ , ⋯ , ℓ ) . \\array{ L & \\overset{diag}{\\hookrightarrow} & P \\coloneqq \\underset{n \\; factors}{\\underbrace{L \\times \\cdots \\times L}} \\\\ \\ell &\\mapsto& (\\ell, \\cdots, \\ell) } \\,. This code “protects against n / 2 − 1 n/2-1 errors” in an evident sense. Much attention in [coding theory](https://ncatlab.org/nlab/show/coding+theory) is instead on the special class of [linear codes](https://ncatlab.org/nlab/show/linear+codes), where L L and P P carry the structure of [vector spaces](https://ncatlab.org/nlab/show/vector+spaces) (necessarily over a [finite field](https://ncatlab.org/nlab/show/finite+field) if they are [finite sets](https://ncatlab.org/nlab/show/finite+sets) of relevance in practice) and where the inclusion L ↪ P L \\hookrightarrow P is a [linear map](https://ncatlab.org/nlab/show/linear+map). ## Examples - [linear code](https://ncatlab.org/nlab/show/linear+code) - [Hamming code](https://ncatlab.org/nlab/show/Hamming+code) - [binary linear code](https://ncatlab.org/nlab/show/binary+linear+code) - [binary Golay code](https://ncatlab.org/nlab/show/binary+Golay+code) - [quantum error correcting code](https://ncatlab.org/nlab/show/quantum+error+correcting+code) - [bit flip code](https://ncatlab.org/nlab/show/bit+flip+code) - [stabilizer code](https://ncatlab.org/nlab/show/stabilizer+code) - [surface code](https://ncatlab.org/nlab/show/surface+code) - [HaPPY code](https://ncatlab.org/nlab/show/HaPPY+code) - [Majorana dimer code](https://ncatlab.org/nlab/show/Majorana+dimer+code) - [toric code](https://ncatlab.org/nlab/show/toric+code) ## References ### General See also the references at [coding theory](https://ncatlab.org/nlab/show/coding+theory) and [linear code](https://ncatlab.org/nlab/show/linear+code). - [Victor V. Albert](https://ncatlab.org/nlab/show/Victor+V.+Albert) et al., [errorcorrectionzoo.org](https://errorcorrectionzoo.org/) - [N. J. A. Sloane](https://ncatlab.org/nlab/show/N.+J.+A.+Sloane), Error-Correcting Codes and Cryptography, The Mathematical Gardner, D. A. Klarner (editor), Prindle, Weber & Schmidt, Boston, MA, 1981, pp. 346-382, Reprinted in “yptologia’’, Vol. 6 (1982), 128-153 and 258-278. An observation on classical codes preconceiving aspects of [holographic tensor network](https://ncatlab.org/nlab/show/holographic+tensor+network) [quantum error correcting codes](https://ncatlab.org/nlab/show/quantum+error+correcting+codes): - [Beni Yoshida](https://ncatlab.org/nlab/show/Beni+Yoshida), Information storage capacity of discrete spin systems, Annals of Physics 338, 134 (2013) ([arXiv:1111.3275](https://arxiv.org/abs/1111.3275)) ### Relation to 2d CFT Construction of chiral [2d SCFTs](https://ncatlab.org/nlab/show/2d+SCFTs) from error-correcting codes: - [Davide Gaiotto](https://ncatlab.org/nlab/show/Davide+Gaiotto), [Theo Johnson-Freyd](https://ncatlab.org/nlab/show/Theo+Johnson-Freyd), Holomorphic SCFTs with small index, Canadian Journal of Mathematics , 74 2 (2022) 573-601 �[arXiv:1811.00589](https://arxiv.org/abs/1811.00589), [doi:10.4153/S0008414X2100002X](https://doi.org/10.4153/S0008414X2100002X)� On their [elliptic genera](https://ncatlab.org/nlab/show/elliptic+genera) - Kohki Kawabata, Shinichiro Yahagi, Elliptic genera from classical error-correcting codes �[arXiv:2308.12592](https://arxiv.org/abs/2308.12592)� Last revised on February 3, 2026 at 11:46:41. See the [history](https://ncatlab.org/nlab/history/error+correcting+code) of this page for a list of all contributions to it. [Edit](https://ncatlab.org/nlab/edit/error+correcting+code)[Discuss](https://nforum.ncatlab.org/discussion/12802/#Item_4)[Previous revision](https://ncatlab.org/nlab/revision/error+correcting+code/12)[Changes from previous revision](https://ncatlab.org/nlab/show/diff/error+correcting+code)[History (12 revisions)](https://ncatlab.org/nlab/history/error+correcting+code) [Cite](https://ncatlab.org/nlab/show/error+correcting+code/cite) [Print](https://ncatlab.org/nlab/print/error+correcting+code) [Source](https://ncatlab.org/nlab/source/error+correcting+code)
Readable Markdown	## Contents - [Idea](https://ncatlab.org/nlab/show/error+correcting+code#idea) - [Examples](https://ncatlab.org/nlab/show/error+correcting+code#examples) - [References](https://ncatlab.org/nlab/show/error+correcting+code#references) - [General](https://ncatlab.org/nlab/show/error+correcting+code#general) - [Relation to 2d CFT](https://ncatlab.org/nlab/show/error+correcting+code#relation_to_2d_cft) ## Idea In [coding theory](https://ncatlab.org/nlab/show/coding+theory), an error correcting code is a means to encode data in a way that is robust against errors ([noise](https://ncatlab.org/nlab/show/noise)). Very broadly, for L L a [finite set](https://ncatlab.org/nlab/show/finite+set) playing the role of a [space of states](https://ncatlab.org/nlab/show/space+of+states) that is to be saved/communicated/analyzed, an error correcting code for L L is an [injection](https://ncatlab.org/nlab/show/injection) L ↪ P L \\overset{\\;\\;\\;}{\\hookrightarrow} P into a larger set. The idea is that noise/errors move the image of L L within P P, but if P P is large enough and the embedding chosen well enough, then a sufficiently small number of errors stays within a small neighbourhood of L L in P P that allows to [retract](https://ncatlab.org/nlab/show/retraction) back to L L. The simplest example is the repetition code, where the inclusion is the [diagonal](https://ncatlab.org/nlab/show/diagonal) on the n n\-fold [Cartesian product](https://ncatlab.org/nlab/show/Cartesian+product) L ↪ diag P ≔ L × ⋯ × L ⏟ n factors ℓ ↦ ( ℓ , ⋯ , ℓ ) . \\array{ L & \\overset{diag}{\\hookrightarrow} & P \\coloneqq \\underset{n \\; factors}{\\underbrace{L \\times \\cdots \\times L}} \\\\ \\ell &\\mapsto& (\\ell, \\cdots, \\ell) } \\,. This code “protects against n / 2 − 1 n/2-1 errors” in an evident sense. Much attention in [coding theory](https://ncatlab.org/nlab/show/coding+theory) is instead on the special class of [linear codes](https://ncatlab.org/nlab/show/linear+codes), where L L and P P carry the structure of [vector spaces](https://ncatlab.org/nlab/show/vector+spaces) (necessarily over a [finite field](https://ncatlab.org/nlab/show/finite+field) if they are [finite sets](https://ncatlab.org/nlab/show/finite+sets) of relevance in practice) and where the inclusion L ↪ P L \\hookrightarrow P is a [linear map](https://ncatlab.org/nlab/show/linear+map). ## Examples - [linear code](https://ncatlab.org/nlab/show/linear+code) - [Hamming code](https://ncatlab.org/nlab/show/Hamming+code) - [binary linear code](https://ncatlab.org/nlab/show/binary+linear+code) - [binary Golay code](https://ncatlab.org/nlab/show/binary+Golay+code) - [quantum error correcting code](https://ncatlab.org/nlab/show/quantum+error+correcting+code) - [bit flip code](https://ncatlab.org/nlab/show/bit+flip+code) - [stabilizer code](https://ncatlab.org/nlab/show/stabilizer+code) - [surface code](https://ncatlab.org/nlab/show/surface+code) - [HaPPY code](https://ncatlab.org/nlab/show/HaPPY+code) - [Majorana dimer code](https://ncatlab.org/nlab/show/Majorana+dimer+code) - [toric code](https://ncatlab.org/nlab/show/toric+code) ## References ### General See also the references at [coding theory](https://ncatlab.org/nlab/show/coding+theory) and [linear code](https://ncatlab.org/nlab/show/linear+code). - [Victor V. Albert](https://ncatlab.org/nlab/show/Victor+V.+Albert) et al., [errorcorrectionzoo.org](https://errorcorrectionzoo.org/) - [N. J. A. Sloane](https://ncatlab.org/nlab/show/N.+J.+A.+Sloane), Error-Correcting Codes and Cryptography, The Mathematical Gardner, D. A. Klarner (editor), Prindle, Weber & Schmidt, Boston, MA, 1981, pp. 346-382, Reprinted in “yptologia’’, Vol. 6 (1982), 128-153 and 258-278. An observation on classical codes preconceiving aspects of [holographic tensor network](https://ncatlab.org/nlab/show/holographic+tensor+network) [quantum error correcting codes](https://ncatlab.org/nlab/show/quantum+error+correcting+codes): - [Beni Yoshida](https://ncatlab.org/nlab/show/Beni+Yoshida), Information storage capacity of discrete spin systems, Annals of Physics 338, 134 (2013) ([arXiv:1111.3275](https://arxiv.org/abs/1111.3275)) ### Relation to 2d CFT Construction of chiral [2d SCFTs](https://ncatlab.org/nlab/show/2d+SCFTs) from error-correcting codes: - [Davide Gaiotto](https://ncatlab.org/nlab/show/Davide+Gaiotto), [Theo Johnson-Freyd](https://ncatlab.org/nlab/show/Theo+Johnson-Freyd), Holomorphic SCFTs with small index, Canadian Journal of Mathematics , 74 2 (2022) 573-601 �[arXiv:1811.00589](https://arxiv.org/abs/1811.00589), [doi:10.4153/S0008414X2100002X](https://doi.org/10.4153/S0008414X2100002X)� On their [elliptic genera](https://ncatlab.org/nlab/show/elliptic+genera) - Kohki Kawabata, Shinichiro Yahagi, Elliptic genera from classical error-correcting codes �[arXiv:2308.12592](https://arxiv.org/abs/2308.12592)� Last revised on February 3, 2026 at 11:46:41. See the [history](https://ncatlab.org/nlab/history/error+correcting+code) of this page for a list of all contributions to it.
Shard	14 (laksa)
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