{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,9]],"date-time":"2026-06-09T19:13:58Z","timestamp":1781032438510,"version":"3.54.1"},"reference-count":56,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2024,3,1]],"date-time":"2024-03-01T00:00:00Z","timestamp":1709251200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,3,1]],"date-time":"2024-03-01T00:00:00Z","timestamp":1709251200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"European Union NextGenerationEU\/PRTR"},{"name":"European Union NextGenerationEU\/PRTR"},{"DOI":"10.13039\/501100001602","name":"Science Foundation Ireland","doi-asserted-by":"crossref","award":["18\/CRT\/6049"],"award-info":[{"award-number":["18\/CRT\/6049"]}],"id":[{"id":"10.13039\/501100001602","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/501100001631","name":"University College Dublin","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100001631","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Quantum Inf Process"],"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We construct new stabilizer quantum error-correcting codes from generalized monomial-Cartesian codes. Our construction uses an explicitly defined twist vector, and we present formulas for the minimum distance and dimension. Generalized monomial-Cartesian codes arise from polynomials in<jats:italic>m<\/jats:italic>variables. When<jats:inline-formula><jats:alternatives><jats:tex-math>$$m=1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>m<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>our codes are MDS, and when<jats:inline-formula><jats:alternatives><jats:tex-math>$$m=2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>m<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and our lower bound for the minimum distance is 3, the codes are at least Hermitian almost MDS. For an infinite family of parameters, when<jats:inline-formula><jats:alternatives><jats:tex-math>$$m=2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>m<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>we prove that our codes beat the Gilbert\u2013Varshamov bound. We also present many examples of our codes that are better than any known code in the literature.<\/jats:p>","DOI":"10.1007\/s11128-024-04297-x","type":"journal-article","created":{"date-parts":[[2024,3,1]],"date-time":"2024-03-01T06:02:31Z","timestamp":1709272951000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["MDS, Hermitian almost MDS, and Gilbert\u2013Varshamov quantum codes from generalized monomial-Cartesian codes"],"prefix":"10.1007","volume":"23","author":[{"given":"Beatriz","family":"Barbero-Lucas","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fernando","family":"Hernando","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Helena","family":"Mart\u00edn-Cruz","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2105-9792","authenticated-orcid":false,"given":"Gary","family":"McGuire","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2024,3,1]]},"reference":[{"issue":"3","key":"4297_CR1","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1109\/TIT.2006.890730","volume":"53","author":"SA Aly","year":"2007","unstructured":"Aly, S.A., Klappenecker, A., Sarvepalli, P.K.: On quantum and classical BCH codes. IEEE Trans. Inf. Theory 53(3), 1183\u20131188 (2007)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"1","key":"4297_CR2","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.ffa.2006.12.004","volume":"14","author":"HE Andersen","year":"2008","unstructured":"Andersen, H.E., Geil, O.: Evaluation codes from order domain theory. Finite Fields their Appl. 14(1), 92\u2013123 (2008)","journal-title":"Finite Fields their Appl."},{"key":"4297_CR3","doi-asserted-by":"crossref","first-page":"778","DOI":"10.1109\/18.841162","volume":"46","author":"A Ashikhmin","year":"2000","unstructured":"Ashikhmin, A., Barg, A., Knill, E., Litsyn, S.: Quantum error-detection I: Statement of the problem. IEEE Trans. Inf. Theory 46, 778\u2013788 (2000)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"4297_CR4","doi-asserted-by":"crossref","first-page":"789","DOI":"10.1109\/18.841163","volume":"46","author":"A Ashikhmin","year":"2000","unstructured":"Ashikhmin, A., Barg, A., Knill, E., Litsyn, S.: Quantum error-detection II: bounds. IEEE Trans. Inf. Theory 46, 789\u2013800 (2000)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"4297_CR5","doi-asserted-by":"crossref","first-page":"3065","DOI":"10.1109\/18.959288","volume":"47","author":"A Ashikhmin","year":"2001","unstructured":"Ashikhmin, A., Knill, E.: Non-binary quantum stabilizer codes. IEEE Trans. Inf. Theory 47, 3065\u20133072 (2001)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"3","key":"4297_CR6","doi-asserted-by":"crossref","first-page":"032311","DOI":"10.1103\/PhysRevA.63.032311","volume":"63","author":"A Ashikhmin","year":"2001","unstructured":"Ashikhmin, A., Litsyn, S., Tsfasman, M.A.: Asymptotically good quantum codes. Phys. Rev. A 63(3), 032311 (2001)","journal-title":"Phys. Rev. A"},{"key":"4297_CR7","doi-asserted-by":"crossref","first-page":"811","DOI":"10.1007\/s10623-021-00846-y","volume":"89","author":"S Ball","year":"2021","unstructured":"Ball, S.: Some constructions of quantum MDS codes. Des. Codes Cryptogr. 89, 811\u2013821 (2021)","journal-title":"Des. Codes Cryptogr."},{"key":"4297_CR8","doi-asserted-by":"crossref","unstructured":"Bhardwaj, S., Goyal, M., Raka, M.: New quantum codes from constacyclic codes over a general non-chain ring. arXiv preprint arXiv:2212.02821, (2022)","DOI":"10.1142\/S1793830923500726"},{"key":"4297_CR9","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1002\/(SICI)1520-6610(2000)8:3<174::AID-JCD3>3.0.CO;2-T","volume":"8","author":"J Bierbrauer","year":"2000","unstructured":"Bierbrauer, J., Edel, Y.: Quantum twisted codes. J. Comb. Designs 8, 174\u2013188 (2000)","journal-title":"J. Comb. Designs"},{"key":"4297_CR10","doi-asserted-by":"crossref","first-page":"S1","DOI":"10.1038\/d41586-023-01692-9","volume":"617","author":"M Brooks","year":"2023","unstructured":"Brooks, M.: Quantum computers: what are they good for? Nature 617, S1\u2013S3 (2023)","journal-title":"Nature"},{"key":"4297_CR11","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1103\/PhysRevLett.78.405","volume":"76","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. 76, 405\u2013409 (1997)","journal-title":"Phys. Rev. Lett."},{"issue":"4","key":"4297_CR12","doi-asserted-by":"crossref","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 $${{\\rm GF}}(4)$$. IEEE Trans. Inf. Theory 44(4), 1369\u20131387 (1998)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"6","key":"4297_CR13","doi-asserted-by":"crossref","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)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"4297_CR14","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1007\/s11128-020-02921-0","volume":"19","author":"M Cao","year":"2020","unstructured":"Cao, M., Cui, J.: Construction of new quantum codes via Hermitian dual-containing matrix-product codes. Quantum Inf. Process. 19, 427 (2020)","journal-title":"Quantum Inf. Process."},{"issue":"7635","key":"4297_CR15","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1038\/541009a","volume":"541","author":"D Castelvecchi","year":"2017","unstructured":"Castelvecchi, D.: Quantum computers ready to leap out of the lab in 2017. Nature 541(7635), 9\u201310 (2017)","journal-title":"Nature"},{"key":"4297_CR16","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1007\/s10623-012-9620-7","volume":"69","author":"G Chen","year":"2013","unstructured":"Chen, G., Li, R.: Ternary self-orthogonal codes of dual distance three and ternary quantum codes of distance three. Des. Codes Cryptogr. 69, 53\u201363 (2013)","journal-title":"Des. Codes Cryptogr."},{"key":"4297_CR17","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-35651-8","volume-title":"Ideals, Varieties, and Algorithms, Undergraduate Texts in Mathematics","author":"D Cox","year":"2007","unstructured":"Cox, D., Little, J., O\u2019Shea, D.: An Introduction to Computational Algebraic Geometry and Commutative Algebra. In: Axler, S., Ribet, K. (eds.) Ideals, Varieties, and Algorithms, Undergraduate Texts in Mathematics. Springer, New York (2007)"},{"key":"4297_CR18","first-page":"271","volume":"92","author":"D Dieks","year":"1982","unstructured":"Dieks, D.: Communication by EPR devices. Phys. Rev. A 92, 271 (1982)","journal-title":"Phys. Rev. A"},{"key":"4297_CR19","doi-asserted-by":"crossref","first-page":"7840","DOI":"10.1109\/TIT.2019.2939114","volume":"65","author":"W Fang","year":"2019","unstructured":"Fang, W., Fu, F.W.: Some new constructions of quantum MDS codes. IEEE Trans. Inf. Theory 65, 7840\u20137847 (2019)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"1","key":"4297_CR20","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1109\/18.179340","volume":"39","author":"G-L 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), 37\u201345 (1993)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"4297_CR21","series-title":"Lecture Notes Series","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1142\/9789812388841_0003","volume-title":"Coding Theory and Cryptology","author":"K Feng","year":"2002","unstructured":"Feng, K.: Quantum error correcting codes. In: Niederreiter, H. (ed.) Coding Theory and Cryptology. Lecture Notes Series, vol. 1, pp. 91\u2013142. National University of Singapore, Institute for Mathematical Sciences (2002)"},{"issue":"12","key":"4297_CR22","doi-asserted-by":"crossref","first-page":"3323","DOI":"10.1109\/TIT.2004.838088","volume":"50","author":"K Feng","year":"2004","unstructured":"Feng, K., Ma, Z.: A finite Gilbert-Varshamov bound for pure stabilizer quantum codes. IEEE Trans. Inf. Theory 50(12), 3323\u20133325 (2004)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"4297_CR23","doi-asserted-by":"crossref","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 Basis. Des. Codes Cryptogr. 13, 147\u2013158 (1998)","journal-title":"Des. Codes Cryptogr."},{"key":"4297_CR24","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/s11128-017-1559-1","volume":"164","author":"C Galindo","year":"2017","unstructured":"Galindo, C., Geil, O., Hernando, F., Ruano, D.: On the distance of stabilizer quantum codes from $$J$$-affine variety codes. Quantum Inf Process. 164, 111 (2017)","journal-title":"Quantum Inf Process."},{"issue":"4","key":"4297_CR25","doi-asserted-by":"crossref","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)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"4297_CR26","doi-asserted-by":"crossref","unstructured":"Galindo, C., Hernando, F., Mart\u00edn-Cruz, H.: Optimal $$(r,\\delta )$$-LRCs from monomial-Cartesian codes and their subfield-subcodes. arXiv preprint arXiv:2205.01485, (2023)","DOI":"10.1007\/s10623-024-01403-z"},{"key":"4297_CR27","doi-asserted-by":"crossref","first-page":"102138","DOI":"10.1016\/j.ffa.2022.102138","volume":"87","author":"C Galindo","year":"2023","unstructured":"Galindo, C., Hernando, F., Mart\u00edn-Cruz, H., Ruano, D.: Stabilizer quantum codes defined by trace-depending polynomials. Finite Fields their Appl. 87, 102138 (2023)","journal-title":"Finite Fields their Appl."},{"key":"4297_CR28","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1016\/j.ffa.2015.07.003","volume":"36","author":"C Galindo","year":"2015","unstructured":"Galindo, C., Hernando, F., Ruano, D.: New quantum codes from evaluation and matrix-product codes. Finite Fields their Appl. 36, 98\u2013120 (2015)","journal-title":"Finite Fields their Appl."},{"issue":"2","key":"4297_CR29","doi-asserted-by":"crossref","first-page":"635","DOI":"10.1109\/18.825832","volume":"46","author":"O Geil","year":"2000","unstructured":"Geil, O., Hoholdt, T.: Footprints or generalized Bezout\u2019s theorem. IEEE Trans. Inf. Theory 46(2), 635\u2013641 (2000)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"4297_CR30","doi-asserted-by":"crossref","unstructured":"Geil, O., H\u00f8holdt, T.: On hyperbolic codes. In: Boztas, S., Shparlinski, I.E. (eds.) Applied Algebra. Algebraic Algorithms and Error-Correcting Codes, volume 2227 of Lecture Notes in Computer Science, pp. 159\u2013171. Springer, Berlin, Germany (2001)","DOI":"10.1007\/3-540-45624-4_17"},{"issue":"3","key":"4297_CR31","doi-asserted-by":"crossref","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)","journal-title":"Phys. Rev. A"},{"issue":"1","key":"4297_CR32","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1142\/S0219749904000079","volume":"2","author":"M Grassl","year":"2004","unstructured":"Grassl, M., Beth, T., R\u00f6tteler, M.: On optimal quantum codes. Int. J. Quantum Inf. 2(1), 55\u201364 (2004)","journal-title":"Int. J. Quantum Inf."},{"key":"4297_CR33","unstructured":"Grassl, M., R\u00f6tteler, M.: Quantum BCH codes. In Proc. X Int. Symp. Theor. Elec. Eng., pages 207\u2013212, (1999)"},{"key":"4297_CR34","first-page":"871","volume-title":"Handbook of Coding Theory","author":"T H\u00f8holdt","year":"1998","unstructured":"H\u00f8holdt, T., van Lint, J.H., Pellikaan, G.R.: Algebraic geometry codes. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, pp. 871\u2013961. Elsevier, Netherlands (1998)"},{"issue":"11","key":"4297_CR35","doi-asserted-by":"crossref","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\u20134914 (2006)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"8","key":"4297_CR36","doi-asserted-by":"crossref","first-page":"2392","DOI":"10.1109\/TIT.2002.800471","volume":"48","author":"A Klappenecker","year":"2002","unstructured":"Klappenecker, A., R\u00f6tteler, M.: Beyond stabilizer codes I: nice error bases. IEEE Trans. Inf. Theory 48(8), 2392\u20132395 (2002)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"4297_CR37","doi-asserted-by":"crossref","unstructured":"Knill, E.: Non-Binary Unitary Error Bases and Quantum Codes. Technical report, Los Alamos National Laboratory, LAUR-96-2717, (1996)","DOI":"10.2172\/373768"},{"issue":"3","key":"4297_CR38","first-page":"609","volume":"56","author":"E Koloto\u011flu","year":"2019","unstructured":"Koloto\u011flu, E., Sar\u0131, M.: Quantum codes with improved minimum distance. Bull. Korean Math. Soc. 56(3), 609\u2013619 (2019)","journal-title":"Bull. Korean Math. Soc."},{"key":"4297_CR39","doi-asserted-by":"crossref","unstructured":"Kong, B., Zheng, X.: Quantum codes from constacyclic codes over $$S_k$$. EPJ Quantum Technol., 10(3), (2023)","DOI":"10.1140\/epjqt\/s40507-023-00160-7"},{"key":"4297_CR40","doi-asserted-by":"crossref","first-page":"042331","DOI":"10.1103\/PhysRevA.80.042331","volume":"80","author":"GG La Guardia","year":"2009","unstructured":"La Guardia, G.G.: Construction of new families of nonbinary quantum codes. Phys. Rev. A 80, 042331 (2009)","journal-title":"Phys. Rev. A"},{"issue":"3","key":"4297_CR41","doi-asserted-by":"crossref","first-page":"1528","DOI":"10.1109\/TIT.2014.2298137","volume":"60","author":"GG La Guardia","year":"2014","unstructured":"La Guardia, G.G.: On the construction of nonbinary quantum BCH codes. IEEE Trans. Inf. Theory 60(3), 1528\u20131535 (2014)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"14","key":"4297_CR42","first-page":"1","volume":"20","author":"H Liu","year":"2021","unstructured":"Liu, H., Liu, X.: Constructions of quantum MDS codes. Quantum Inf. Process. 20(14), 1\u20133 (2021)","journal-title":"Quantum Inf. Process."},{"key":"4297_CR43","doi-asserted-by":"crossref","first-page":"579","DOI":"10.1007\/s12095-017-0242-9","volume":"10","author":"X Liu","year":"2018","unstructured":"Liu, X., Dinh, H.Q., Liu, H., Yu, L.: On new quantum codes from matrix product codes. Cryptogr. Commun. 10, 579\u2013589 (2018)","journal-title":"Cryptogr. Commun."},{"key":"4297_CR44","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1007\/s11128-023-04211-x","volume":"23","author":"G Luo","year":"2024","unstructured":"Luo, G., Ezerman, M.F., Grassl, M., Ling, S.: Constructing quantum error-correcting codes that require a variable amount of entanglement. Quantum Inf. Process. 23, 4 (2024)","journal-title":"Quantum Inf. Process."},{"key":"4297_CR45","doi-asserted-by":"crossref","first-page":"1673","DOI":"10.1007\/s10623-020-00726-x","volume":"88","author":"HH L\u00f3pez","year":"2020","unstructured":"L\u00f3pez, H.H., Matthews, G.L., Soprunov, I.: Monomial-Cartesian codes and their duals, with applications to LCD codes, quantum codes, and locally recoverable codes. Des. Codes Cryptogr. 88, 1673\u20131685 (2020)","journal-title":"Des. Codes Cryptogr."},{"key":"4297_CR46","doi-asserted-by":"crossref","first-page":"1367","DOI":"10.1007\/s10623-021-00872-w","volume":"89","author":"HH L\u00f3pez","year":"2021","unstructured":"L\u00f3pez, H.H., Soprunov, I., Villarreal, R.H.: The dual of an evaluation code. Des. Codes Cryptogr. 89, 1367\u20131403 (2021)","journal-title":"Des. Codes Cryptogr."},{"issue":"10","key":"4297_CR47","first-page":"1878","volume":"83","author":"R Matsumoto","year":"2000","unstructured":"Matsumoto, R., Uyematsu, T.: Constructing quantum error-correcting codes for $$p^m$$-state systems from classical error-correcting codes. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. 83(10), 1878\u20131883 (2000)","journal-title":"IEICE Trans. Fundam. Electron. Commun. Comput. Sci."},{"issue":"4","key":"4297_CR48","doi-asserted-by":"crossref","first-page":"1388","DOI":"10.1109\/18.681316","volume":"44","author":"EM Rains","year":"1998","unstructured":"Rains, E.M.: Quantum weight enumerators. IEEE Trans. Inf. Theory 44(4), 1388\u20131394 (1998)","journal-title":"IEEE Trans. Inf. Theory"},{"issue":"4","key":"4297_CR49","doi-asserted-by":"crossref","first-page":"2493","DOI":"10.1103\/PhysRevA.52.R2493","volume":"52","author":"PW Shor","year":"1995","unstructured":"Shor, P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52(4), 2493\u20132496 (1995)","journal-title":"Phys. Rev. A"},{"issue":"5","key":"4297_CR50","doi-asserted-by":"crossref","first-page":"1484","DOI":"10.1137\/S0097539795293172","volume":"26","author":"PW Shor","year":"1997","unstructured":"Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. Comput. 26(5), 1484\u20131509 (1997)","journal-title":"SIAM J. Comput."},{"issue":"226","key":"4297_CR51","first-page":"1","volume":"19","author":"H Song","year":"2020","unstructured":"Song, H., Li, R., Liu, Y., Guo, G.: New quantum codes from matrix-product codes over small fields. Quantum Inf. Process. 19(226), 1\u201322 (2020)","journal-title":"Quantum Inf. Process."},{"key":"4297_CR52","doi-asserted-by":"crossref","first-page":"2551","DOI":"10.1098\/rspa.1996.0136","volume":"452","author":"A Steane","year":"1996","unstructured":"Steane, A.: Multiple-particle interference and quantum error correction. Proc. R. Soc. Lond. A 452, 2551\u20132577 (1996)","journal-title":"Proc. R. Soc. Lond. A"},{"issue":"6","key":"4297_CR53","doi-asserted-by":"crossref","first-page":"4741","DOI":"10.1103\/PhysRevA.54.4741","volume":"54","author":"AM Steane","year":"1996","unstructured":"Steane, A.M.: Simple quantum error-correcting codes. Phys. Rev. A 54(6), 4741\u20134751 (1996)","journal-title":"Phys. Rev. A"},{"key":"4297_CR54","doi-asserted-by":"crossref","unstructured":"Wan, R., Zhu, S.: New Quantum MDS codes from Hermitian self-orthogonal generalized Reed-Solomon codes. arXiv preprint arXiv:2302.06169, (2023)","DOI":"10.1007\/s12095-024-00752-9"},{"issue":"122","key":"4297_CR55","first-page":"1","volume":"20","author":"Y Wang","year":"2021","unstructured":"Wang, Y., Kai, X., Sun, Z., Zhu, S.: Quantum codes from Hermitian dual-containing constacyclic codes over $${{\\mathbb{F} }}_{q^{2}}+{v}{{\\mathbb{F} }}_{q^{2}}$$. Quantum Inf. Process. 20(122), 1\u201317 (2021)","journal-title":"Quantum Inf. Process."},{"key":"4297_CR56","doi-asserted-by":"crossref","first-page":"802","DOI":"10.1038\/299802a0","volume":"299","author":"WK Wootters","year":"1982","unstructured":"Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299, 802\u2013803 (1982)","journal-title":"Nature"}],"container-title":["Quantum Information Processing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-024-04297-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11128-024-04297-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11128-024-04297-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,13]],"date-time":"2024-11-13T17:02:49Z","timestamp":1731517369000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11128-024-04297-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,3,1]]},"references-count":56,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,3]]}},"alternative-id":["4297"],"URL":"https:\/\/doi.org\/10.1007\/s11128-024-04297-x","relation":{},"ISSN":["1573-1332"],"issn-type":[{"value":"1573-1332","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,3,1]]},"assertion":[{"value":"30 July 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"31 January 2024","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 March 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"We declare no conflicts of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"86"}}