{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T09:02:02Z","timestamp":1785142922564,"version":"3.55.0"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"7","license":[{"start":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T00:00:00Z","timestamp":1783036800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T00:00:00Z","timestamp":1783036800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100008530","name":"European Regional Development Fund","doi-asserted-by":"publisher","award":["PID2022-138906NB-C22"],"award-info":[{"award-number":["PID2022-138906NB-C22"]}],"id":[{"id":"10.13039\/501100008530","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100014440","name":"Ministerio de Ciencia, Innovaci\u00f3n y Universidades","doi-asserted-by":"publisher","award":["MICIU\/AEI\/-10.13039\/-501100011033"],"award-info":[{"award-number":["MICIU\/AEI\/-10.13039\/-501100011033"]}],"id":[{"id":"10.13039\/100014440","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["23K10980"],"award-info":[{"award-number":["23K10980"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Inf Process"],"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Existing literature provides several bounds for quantum local recovery, which essentially consider the number of message qudits, the distance, the length, and the locality of the involved codes. We give a family of\n                    <jats:italic>J<\/jats:italic>\n                    -affine variety codes that result in impure CSS codes. These quantum codes exceed several of the above mentioned bounds that apply to pure quantum locally recoverable codes. We also discuss a connection between bounds on quantum local recovery and on weight-constrained stabilizer codes.\n                  <\/jats:p>","DOI":"10.1007\/s11128-026-05259-1","type":"journal-article","created":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T06:28:04Z","timestamp":1783060084000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Impure codes exceeding the pure bounds for quantum local recovery"],"prefix":"10.1007","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3908-4462","authenticated-orcid":false,"given":"Carlos","family":"Galindo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9758-2152","authenticated-orcid":false,"given":"Fernando","family":"Hernando","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6379-6902","authenticated-orcid":false,"given":"Helena","family":"Mart\u00edn-Cruz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5085-8879","authenticated-orcid":false,"given":"Ryutaroh","family":"Matsumoto","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,7,3]]},"reference":[{"issue":"7","key":"5259_CR1","doi-asserted-by":"publisher","first-page":"3065","DOI":"10.1109\/18.959288","volume":"47","author":"A Ashikhmin","year":"2001","unstructured":"Ashikhmin, A., Knill, E.: Nonbinary quantum stabilizer codes. IEEE Trans. Inf. Theory 47(7), 3065\u20133072 (2001). https:\/\/doi.org\/10.1109\/18.959288","journal-title":"IEEE Trans. Inf. Theory"},{"key":"5259_CR2","doi-asserted-by":"publisher","first-page":"3217","DOI":"10.1103\/PhysRevLett.78.3217","volume":"78","author":"CH Bennett","year":"1997","unstructured":"Bennett, C.H., DiVincenzo, D.P., Smolin, J.A.: Capacities of quantum erasure channels. Phys. Rev. Lett. 78, 3217\u20133220 (1997). https:\/\/doi.org\/10.1103\/PhysRevLett.78.3217","journal-title":"Phys. Rev. Lett."},{"issue":"6","key":"5259_CR3","doi-asserted-by":"publisher","first-page":"3073","DOI":"10.1109\/TIT.2014.2313559","volume":"60","author":"TA Brun","year":"2014","unstructured":"Brun, T.A., Devetak, I., Hsieh, M.H.: Catalytic quantum error correction. IEEE Trans. Inf. Theory 60(6), 3073\u20133089 (2014). https:\/\/doi.org\/10.1109\/TIT.2014.2313559","journal-title":"IEEE Trans. Inf. Theory"},{"key":"5259_CR4","unstructured":"Bu, K., Gu, W., Li, X.: Quantum locally recoverable code with intersecting recovery sets (2025). arXiv:2501.10354"},{"issue":"3","key":"5259_CR5","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1103\/PhysRevLett.78.405","volume":"78","author":"AR Calderbank","year":"1997","unstructured":"Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction and orthogonal geometry. Phys. Rev. Lett. 78(3), 405\u2013408 (1997). https:\/\/doi.org\/10.1103\/PhysRevLett.78.405","journal-title":"Phys. Rev. Lett."},{"issue":"4","key":"5259_CR6","doi-asserted-by":"publisher","first-page":"1369","DOI":"10.1109\/18.681315","volume":"44","author":"AR Calderbank","year":"1998","unstructured":"Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44(4), 1369\u20131387 (1998). https:\/\/doi.org\/10.1109\/18.681315","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"2","key":"5259_CR7","doi-asserted-by":"publisher","first-page":"1098","DOI":"10.1103\/PhysRevA.54.1098","volume":"54","author":"AR Calderbank","year":"1996","unstructured":"Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54(2), 1098\u20131105 (1996). https:\/\/doi.org\/10.1103\/PhysRevA.54.1098","journal-title":"Phys. Rev. A"},{"issue":"6","key":"5259_CR8","doi-asserted-by":"publisher","first-page":"3664","DOI":"10.1109\/TIT.2020.3047624","volume":"67","author":"E Camps","year":"2021","unstructured":"Camps, E., L\u00f3pez, H.H., Matthews, G.L., Sarmiento, E.: Polar decreasing monomial-Cartesian codes. IEEE Trans. Inf. Theory 67(6), 3664\u20133674 (2021). https:\/\/doi.org\/10.1109\/TIT.2020.3047624","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"6","key":"5259_CR9","doi-asserted-by":"publisher","first-page":"3885","DOI":"10.1109\/TIT.2026.3681560","volume":"72","author":"M Cao","year":"2026","unstructured":"Cao, M., Zhou, K.: An open problem on NSC quasi-unitary matrices and its application to quantum codes. IEEE Trans. Inf. Theory 72(6), 3885\u20133898 (2026). https:\/\/doi.org\/10.1109\/TIT.2026.3681560","journal-title":"IEEE Trans. Inf. Theory"},{"key":"5259_CR10","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2026.3691784","author":"M Cao","year":"2026","unstructured":"Cao, M., Zhou, K.: Optimal quantum $$(r,\\delta )$$-locally repairable codes from matrix-product codes. IEEE Trans. Inf. Theory (2026). https:\/\/doi.org\/10.1109\/TIT.2026.3691784","journal-title":"IEEE Trans. Inf. Theory"},{"key":"5259_CR11","doi-asserted-by":"publisher","unstructured":"Cox, D.A., Little, J., O\u2019Shea, D.: Ideals, varieties, and algorithms, 5th edn. Undergraduate Texts in Mathematics. Springer, Cham, Switzerland (2025). https:\/\/doi.org\/10.1007\/978-3-031-91841-4","DOI":"10.1007\/978-3-031-91841-4"},{"issue":"3","key":"5259_CR12","doi-asserted-by":"publisher","first-page":"1067","DOI":"10.1109\/18.256518","volume":"39","author":"IM Duursma","year":"1993","unstructured":"Duursma, I.M.: Majority coset decoding. IEEE Trans. Inf. Theory 39(3), 1067\u20131070 (1993). https:\/\/doi.org\/10.1109\/18.256518","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"1","key":"5259_CR13","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1016\/j.ffa.2009.11.006","volume":"16","author":"IM Duursma","year":"2010","unstructured":"Duursma, I.M., Park, S.: Coset bounds for algebraic geometric codes. Finite Fields Appl. 16(1), 36\u201355 (2010). https:\/\/doi.org\/10.1016\/j.ffa.2009.11.006","journal-title":"Finite Fields Appl."},{"issue":"1","key":"5259_CR14","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1109\/18.179340","volume":"39","author":"GL Feng","year":"1993","unstructured":"Feng, G.L., Rao, T.R.N.: Decoding algebraic geometric codes up to the designed minimum distance. IEEE Trans. Inf. Theory 39(1), 36\u201347 (1993). https:\/\/doi.org\/10.1109\/18.179340","journal-title":"IEEE Trans. Inf. Theory"},{"key":"5259_CR15","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1023\/A:1008274212057","volume":"13","author":"J Fitzgerald","year":"1998","unstructured":"Fitzgerald, J., Lax, R.F.: Decoding affine variety codes using Gr\u00f6bner bases. Des. Codes Cryptogr. 13, 147\u2013158 (1998). https:\/\/doi.org\/10.1023\/A:1008274212057","journal-title":"Des. Codes Cryptogr."},{"issue":"4","key":"5259_CR16","doi-asserted-by":"publisher","first-page":"2444","DOI":"10.1109\/TIT.2017.2755682","volume":"64","author":"C Galindo","year":"2018","unstructured":"Galindo, C., Geil, O., Hernando, F., Ruano, D.: Improved constructions of nested code pairs. IEEE Trans. Inf. Theory 64(4), 2444\u20132459 (2018). https:\/\/doi.org\/10.1109\/TIT.2017.2755682","journal-title":"IEEE Trans. Inf. Theory"},{"key":"5259_CR17","doi-asserted-by":"publisher","first-page":"2549","DOI":"10.1007\/s10623-024-01403-z","volume":"92","author":"C Galindo","year":"2024","unstructured":"Galindo, C., Hernando, F., Mart\u00edn-Cruz, H.: Optimal $$(r,\\delta )$$-LRCs from monomial-Cartesian codes and their subfield-subcodes. Des. Codes Crypt. 92, 2549\u20132586 (2024). https:\/\/doi.org\/10.1007\/s10623-024-01403-z","journal-title":"Des. Codes Crypt."},{"key":"5259_CR18","doi-asserted-by":"publisher","first-page":"102785","DOI":"10.1016\/j.ffa.2025.102785","volume":"111","author":"C Galindo","year":"2026","unstructured":"Galindo, C., Hernando, F., Mart\u00edn-Cruz, H., Matsumoto, R.: Quantum $$(r,\\delta )$$-locally recoverable codes. Finite Fields Appl. 111, 102785 (2026). https:\/\/doi.org\/10.1016\/j.ffa.2025.102785","journal-title":"Finite Fields Appl."},{"key":"5259_CR19","doi-asserted-by":"crossref","unstructured":"Galindo, C., Hernando, F., Matsumoto, R.: Quantum $$(r,\\delta )$$-locally recoverable BCH and homothetic-BCH codes (2026). arXiv:2601.22567","DOI":"10.1016\/j.ffa.2025.102785"},{"key":"5259_CR20","doi-asserted-by":"publisher","first-page":"3211","DOI":"10.1007\/s11128-015-1057-2","volume":"14","author":"C Galindo","year":"2015","unstructured":"Galindo, C., Hernando, F., Ruano, D.: Stabilizer quantum codes from $$J$$-affine variety codes and a new Steane-like enlargement. Quantum Inf. Process. 14, 3211\u20133231 (2015). https:\/\/doi.org\/10.1007\/s11128-015-1057-2","journal-title":"Quantum Inf. Process."},{"key":"5259_CR21","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1016\/j.ffa.2013.03.005","volume":"23","author":"O Geil","year":"2013","unstructured":"Geil, O., Matsumoto, R., Ruano, D.: Feng-Rao decoding of primary codes. Finite Fields Appl. 23, 35\u201352 (2013). https:\/\/doi.org\/10.1016\/j.ffa.2013.03.005","journal-title":"Finite Fields Appl."},{"key":"5259_CR22","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1007\/s10623-012-9680-8","volume":"66","author":"O Geil","year":"2013","unstructured":"Geil, O., Thomsen, C.: Weighted Reed-Muller codes revisited. Des. Codes Cryptogr. 66, 195\u2013220 (2013). https:\/\/doi.org\/10.1007\/s10623-012-9680-8","journal-title":"Des. Codes Cryptogr."},{"key":"5259_CR23","unstructured":"Golowich, L., Guruswami, V.: Quantum locally recoverable codes (2023). arXiv:2311.08653v1"},{"key":"5259_CR24","doi-asserted-by":"publisher","unstructured":"Golowich, L., Guruswami, V.: Quantum locally recoverable codes. In: Proceeding 2025 ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 5512\u20135522 (2025). https:\/\/doi.org\/10.1137\/1.9781611978322.188","DOI":"10.1137\/1.9781611978322.188"},{"issue":"11","key":"5259_CR25","doi-asserted-by":"publisher","first-page":"6925","DOI":"10.1109\/TIT.2012.2208937","volume":"58","author":"P Gopalan","year":"2012","unstructured":"Gopalan, P., Huang, C., Simitci, H., Yekhanin, S.: On the locality of codeword symbols. IEEE Trans. Inf. Theory 58(11), 6925\u20136934 (2012). https:\/\/doi.org\/10.1109\/TIT.2012.2208937","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"3","key":"5259_CR26","doi-asserted-by":"publisher","first-page":"1862","DOI":"10.1103\/PhysRevA.54.1862","volume":"54","author":"D Gottesman","year":"1996","unstructured":"Gottesman, D.: Class of quantum error-correcting codes saturating the quantum Hamming bound. Phys. Rev. A 54(3), 1862\u20131868 (1996). https:\/\/doi.org\/10.1103\/PhysRevA.54.1862","journal-title":"Phys. Rev. A"},{"key":"5259_CR27","doi-asserted-by":"publisher","first-page":"L020601","DOI":"10.1103\/PhysRevA.103.L020601","volume":"103","author":"M Grassl","year":"2021","unstructured":"Grassl, M.: Entanglement-assisted quantum communication beating the quantum Singleton bound. Phys. Rev. A 103, L020601 (2021). https:\/\/doi.org\/10.1103\/PhysRevA.103.L020601","journal-title":"Phys. Rev. A"},{"key":"5259_CR28","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1103\/PhysRevA.56.33","volume":"56","author":"M Grassl","year":"1997","unstructured":"Grassl, M., Beth, T., Pellizzari, T.: Codes for the quantum erasure channel. Phys. Rev. A 56, 33\u201338 (1997). https:\/\/doi.org\/10.1103\/PhysRevA.56.33","journal-title":"Phys. Rev. A"},{"key":"5259_CR29","doi-asserted-by":"publisher","first-page":"010333","DOI":"10.1103\/PRXQuantum.5.010333","volume":"5","author":"S Heu\u00dfen","year":"2024","unstructured":"Heu\u00dfen, S., Locher, D.F., M\u00fcller, M.: Measurement-free fault-tolerant quantum error correction in near-term devices. PRX Quantum 5, 010333 (2024). https:\/\/doi.org\/10.1103\/PRXQuantum.5.010333","journal-title":"PRX Quantum"},{"key":"5259_CR30","doi-asserted-by":"publisher","first-page":"020358","DOI":"10.1103\/PRXQuantum.4.020358","volume":"4","author":"M Kang","year":"2023","unstructured":"Kang, M., Campbell, W.C., Brown, K.R.: Quantum error correction with metastable states of trapped ions using erasure conversion. PRX Quantum 4, 020358 (2023). https:\/\/doi.org\/10.1103\/PRXQuantum.4.020358","journal-title":"PRX Quantum"},{"issue":"11","key":"5259_CR31","doi-asserted-by":"publisher","first-page":"4892","DOI":"10.1109\/TIT.2006.883612","volume":"52","author":"A Ketkar","year":"2006","unstructured":"Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inf. Theory 52(11), 4892\u20134924 (2006). https:\/\/doi.org\/10.1109\/TIT.2006.883612","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"2","key":"5259_CR32","doi-asserted-by":"publisher","first-page":"900","DOI":"10.1103\/PhysRevA.55.900","volume":"55","author":"E Knill","year":"1997","unstructured":"Knill, E., Laflamme, R.: Theory of quantum error-correcting codes. Phys. Rev. A 55(2), 900\u2013911 (1997). https:\/\/doi.org\/10.1103\/PhysRevA.55.900","journal-title":"Phys. Rev. A"},{"key":"5259_CR33","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.2026.3697209","author":"Y Li","year":"2026","unstructured":"Li, Y., Li, S., Lao, H., Luo, G., Ling, S.: On optimal quantum LRCs from the Hermitian construction and $$t$$-designs. IEEE Trans. Inf. Theory (2026). https:\/\/doi.org\/10.1109\/TIT.2026.3697209","journal-title":"IEEE Trans. Inf. Theory"},{"key":"5259_CR34","unstructured":"Li, Y., Li, S., Luo, G., Ling, S.: Improved bounds and optimal constructions of pure quantum locally recoverable codes (2025). arXiv:2512.07256"},{"issue":"3","key":"5259_CR35","doi-asserted-by":"publisher","first-page":"1794","DOI":"10.1109\/TIT.2025.3533494","volume":"71","author":"G Luo","year":"2025","unstructured":"Luo, G., Chen, B., Ezerman, M.F., Ling, S.: Bounds and constructions of quantum locally recoverable codes from quantum CSS codes. IEEE Trans. Inf. Theory 71(3), 1794\u20131802 (2025). https:\/\/doi.org\/10.1109\/TIT.2025.3533494","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"3","key":"5259_CR36","doi-asserted-by":"publisher","first-page":"1222","DOI":"10.1109\/TIT.2004.842763","volume":"51","author":"Y Luo","year":"2005","unstructured":"Luo, Y., Mitrpant, C., Han Vinck, A.J., Chen, K.: Some new characters on the wire-tap channel of type II. IEEE Trans. Inf. Theory 51(3), 1222\u20131229 (2005). https:\/\/doi.org\/10.1109\/TIT.2004.842763","journal-title":"IEEE Trans. Inf. Theory"},{"key":"5259_CR37","doi-asserted-by":"publisher","first-page":"062426","DOI":"10.1103\/PhysRevA.108.062426","volume":"108","author":"MA Perlin","year":"2023","unstructured":"Perlin, M.A., Premakumar, V.N., Wang, J., Saffman, M., Joynt, R.: Fault-tolerant measurement-free quantum error correction with multiqubit gates. Phys. Rev. A 108, 062426 (2023). https:\/\/doi.org\/10.1103\/PhysRevA.108.062426","journal-title":"Phys. Rev. A"},{"key":"5259_CR38","first-page":"3","volume-title":"Handbook of Coding Theory","author":"VS Pless","year":"1998","unstructured":"Pless, V.S., Huffman, W.C., Brualdi, R.A.: An introduction to algebraic codes. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, pp. 3\u2013139. Elsevier, Amsterdam (1998)"},{"key":"5259_CR39","doi-asserted-by":"publisher","unstructured":"Prakash, N., Kamath, G.M., Lalitha, V., Kumar, P.V.: Optimal linear codes with a local-error-correction property. In: 2012 IEEE International Symposium on Information Theory Proceedings, pp. 2776\u20132780 (2012). https:\/\/doi.org\/10.1109\/ISIT.2012.6284028","DOI":"10.1109\/ISIT.2012.6284028"},{"key":"5259_CR40","doi-asserted-by":"publisher","first-page":"100","DOI":"10.1109\/JSAIT.2025.3567480","volume":"6","author":"S Sharma","year":"2025","unstructured":"Sharma, S., Ramkumar, V., Tamo, I.: Quantum locally recoverable codes via good polynomials. IEEE J. Sel. Areas Inf. Theory 6, 100\u2013110 (2025). https:\/\/doi.org\/10.1109\/JSAIT.2025.3567480","journal-title":"IEEE J. Sel. Areas Inf. Theory"},{"issue":"1954","key":"5259_CR41","doi-asserted-by":"publisher","first-page":"2551","DOI":"10.1098\/rspa.1996.0136","volume":"452","author":"AM Steane","year":"1996","unstructured":"Steane, A.M.: Multiple particle interference and quantum error correction. Proc. R. Soc. London Ser. A 452(1954), 2551\u20132577 (1996). https:\/\/doi.org\/10.1098\/rspa.1996.0136","journal-title":"Proc. R. Soc. London Ser. A"},{"key":"5259_CR42","doi-asserted-by":"publisher","first-page":"043253","DOI":"10.1103\/PhysRevResearch.6.043253","volume":"6","author":"S Veroni","year":"2024","unstructured":"Veroni, S., M\u00fcller, M., Giudice, G.: Optimized measurement-free and fault-tolerant quantum error correction for neutral atoms. Phys. Rev. Res. 6, 043253 (2024). https:\/\/doi.org\/10.1103\/PhysRevResearch.6.043253","journal-title":"Phys. Rev. Res."},{"key":"5259_CR43","unstructured":"Wang, L., Liu, A.Z., Li, R., Kubica, A., Gu, S.: Check-weight-constrained quantum codes: bounds and examples (2026). arXiv:2601.15446"},{"key":"5259_CR44","unstructured":"Wei, F., Han, Z., He, A.Y., Li, Z., Liu, Z.-W.: Theory of low-weight quantum codes (2026). arXiv:2601.19848"},{"key":"5259_CR45","doi-asserted-by":"publisher","first-page":"4657","DOI":"10.1038\/s41467-022-32094-6","volume":"13","author":"Y Wu","year":"2022","unstructured":"Wu, Y., Kolkowitz, S., Puri, S., Thompson, J.D.: Erasure conversion for fault-tolerant quantum computing in alkaline earth Rydberg atom arrays. Nat. Commun. 13, 4657 (2022). https:\/\/doi.org\/10.1038\/s41467-022-32094-6","journal-title":"Nat. Commun."},{"key":"5259_CR46","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1007\/s10773-025-05943-5","volume":"64","author":"D Xie","year":"2025","unstructured":"Xie, D., Zhu, S., Sun, Z.: Two families of optimal quantum locally recoverable codes. Int. J. Theor. Phys. 64, 86 (2025). https:\/\/doi.org\/10.1007\/s10773-025-05943-5","journal-title":"Int. J. Theor. Phys."},{"key":"5259_CR47","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcta.2026.106212","volume":"223","author":"K Zhou","year":"2026","unstructured":"Zhou, K., Cao, M.: Optimal quantum $$(r,\\delta )$$-locally repairable codes via classical ones. J. Combinatorial Theory A 223, 106212 (2026). https:\/\/doi.org\/10.1016\/j.jcta.2026.106212","journal-title":"J. Combinatorial Theory A"},{"key":"5259_CR48","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1038\/s41586-025-09543-5","volume":"646","author":"H Zhou","year":"2025","unstructured":"Zhou, H., Zhao, C., Cain, M.: Low-overhead transversal fault tolerance for universal quantum computation. Nature 646, 303\u2013308 (2025). https:\/\/doi.org\/10.1038\/s41586-025-09543-5","journal-title":"Nature"}],"container-title":["Quantum Information Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-026-05259-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11128-026-05259-1","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-026-05259-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,7,27]],"date-time":"2026-07-27T08:11:22Z","timestamp":1785139882000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11128-026-05259-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7,3]]},"references-count":48,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2026,7]]}},"alternative-id":["5259"],"URL":"https:\/\/doi.org\/10.1007\/s11128-026-05259-1","relation":{},"ISSN":["1573-1332"],"issn-type":[{"value":"1573-1332","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,7,3]]},"assertion":[{"value":"6 April 2026","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 June 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 July 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 July 2026","order":5,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Update","order":6,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The structuring of Ref. 47 has been corrected","order":7,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"The authors declare no competing interests.","order":1,"name":"Ethics","label":"Competing interests","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Initial proofs of Lemma\n                      \n                      , and Theorems\n                      \n                      ,\n                      \n                      , and\n                      \n                      were obtained with the help of AI (Google Gemini 3.1). They were completely rewritten by the authors, and no AI-generated texts or equations remain in this paper.","order":2,"name":"Ethics","label":"Use of AI","group":{"name":"EthicsHeading","label":"Declarations"}}],"article-number":"241"}}