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A CSS-T code depends on a CSS-T pair, which is a pair of binary codes $$(C_1, C_2)$$<\/jats:tex-math>\n \n (<\/mml:mo>\n \n C<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n C<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> such that $$C_1$$<\/jats:tex-math>\n \n C<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> contains $$C_2$$<\/jats:tex-math>\n \n C<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, $$C_2$$<\/jats:tex-math>\n \n C<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is even, and the shortening of the dual of $$C_1$$<\/jats:tex-math>\n \n C<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with respect to the support of each codeword of $$C_2$$<\/jats:tex-math>\n \n C<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is self-dual. In this paper, we give new conditions to guarantee that a pair of binary codes $$(C_1, C_2)$$<\/jats:tex-math>\n \n (<\/mml:mo>\n \n C<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n C<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is a CSS-T pair. We define the poset of CSS-T pairs and determine the minimal and maximal elements of the poset. We provide a propagation rule for nondegenerate CSS-T codes. We apply some main results to Reed\u2013Muller, cyclic and extended cyclic codes. We characterize CSS-T pairs of cyclic codes in terms of the defining cyclotomic cosets. We find cyclic and extended cyclic codes to obtain quantum codes with better parameters than those in the literature.\n<\/jats:p>","DOI":"10.1007\/s11128-024-04427-5","type":"journal-article","created":{"date-parts":[[2024,6,6]],"date-time":"2024-06-06T08:02:21Z","timestamp":1717660941000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["An algebraic characterization of binary CSS-T codes and cyclic CSS-T codes for quantum fault tolerance"],"prefix":"10.1007","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9662-8051","authenticated-orcid":false,"given":"Eduardo","family":"Camps-Moreno","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9832-7145","authenticated-orcid":false,"given":"Hiram H.","family":"L\u00f3pez","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8977-8171","authenticated-orcid":false,"given":"Gretchen L.","family":"Matthews","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7304-0087","authenticated-orcid":false,"given":"Diego","family":"Ruano","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0944-7584","authenticated-orcid":false,"given":"Rodrigo","family":"San-Jos\u00e9","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0122-7699","authenticated-orcid":false,"given":"Ivan","family":"Soprunov","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,6,6]]},"reference":[{"key":"4427_CR1","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.113.080501","volume":"113","author":"JT Anderson","year":"2014","unstructured":"Anderson, J.T., Duclos-Cianci, G., Poulin, D.: Fault-tolerant conversion between the Steane and Reed\u2013Muller quantum codes. 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