{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,23]],"date-time":"2026-04-23T02:57:54Z","timestamp":1776913074247,"version":"3.51.2"},"reference-count":46,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2022,7,19]],"date-time":"2022-07-19T00:00:00Z","timestamp":1658188800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11971004"],"award-info":[{"award-number":["11971004"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>By using difference schemes, orthogonal partitions and a replacement method, some new methods to construct pure quantum error-correcting codes are provided from orthogonal arrays. As an application of these methods, we construct several infinite series of quantum error-correcting codes including some optimal ones. Compared with the existing binary quantum codes, more new codes can be constructed, which have a lower number of terms (i.e., the number of computational basis states) for each of their basis states.<\/jats:p>","DOI":"10.3390\/e24071000","type":"journal-article","created":{"date-parts":[[2022,7,19]],"date-time":"2022-07-19T23:10:22Z","timestamp":1658272222000},"page":"1000","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Construction of Binary Quantum Error-Correcting Codes from Orthogonal Array"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9000-802X","authenticated-orcid":false,"given":"Shanqi","family":"Pang","sequence":"first","affiliation":[{"name":"College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China"}]},{"given":"Hanxiao","family":"Xu","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China"}]},{"given":"Mengqian","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,7,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2493","DOI":"10.1103\/PhysRevA.52.R2493","article-title":"Scheme for reducing decoherence in quantum computer memory","volume":"52","author":"Shor","year":"1995","journal-title":"Phys. Rev. A"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1369","DOI":"10.1109\/18.681315","article-title":"Quantum error correction via codes over GF(4)","volume":"44","author":"Calderbank","year":"1998","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3323","DOI":"10.1109\/TIT.2004.838088","article-title":"A finite Gilbert-Varshamov bound for pure stabilizer quantum codes","volume":"50","author":"Feng","year":"2004","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1331","DOI":"10.1109\/TIT.2004.828149","article-title":"Binary construction of quantum codes of minimum distance three and four","volume":"50","author":"Li","year":"2004","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2007","DOI":"10.1090\/S0002-9947-07-04242-0","article-title":"A new construction of quantum error-correcting codes","volume":"360","author":"Feng","year":"2008","journal-title":"Trans. Amer. Math. Soc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"7180","DOI":"10.1109\/TIT.2011.2165149","article-title":"High Performance Single-Error-Correcting Quantum Codes for Amplitude Damping","volume":"57","author":"Shor","year":"2011","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_7","unstructured":"Movassagh, R., and Ouyang, Y. (2020). Constructing quantum codes from any classical code and their embedding in ground space of local hamiltonians. arXiv."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2921","DOI":"10.1109\/TIT.2019.2956142","article-title":"Permutation-invariant constant-excitation quantum codes for amplitude damping","volume":"66","author":"Ouyang","year":"2019","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"062317","DOI":"10.1103\/PhysRevA.90.062317","article-title":"Permutation-invariant quantum codes","volume":"90","author":"Ouyang","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.laa.2017.06.031","article-title":"Permutation-invariant qudit codes from polynomials","volume":"532","author":"Ouyang","year":"2017","journal-title":"Linear Algebra Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"194","DOI":"10.1103\/PhysRevLett.85.194","article-title":"Pauli Exchange Errors in Quantum Computation","volume":"85","author":"Ruskai","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","unstructured":"Grassl, M. (2022, June 01). Bounds on the Minimum Distance of Additive Quantum Codes. Available online: http:\/\/www.codetables.de."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"012306","DOI":"10.1103\/PhysRevA.78.012306","article-title":"Graphical nonbinary quantum error-correcting codes","volume":"78","author":"Hu","year":"2008","journal-title":"Phys. Rev. A"},{"key":"ref_14","unstructured":"Nebe, G., Rains, E.M., and Sloane, N.J.A. (2006). Self-Dual Codes and Invariant Theory, Springer."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"032316","DOI":"10.1103\/PhysRevA.92.032316","article-title":"Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices","volume":"92","author":"Goyeneche","year":"2015","journal-title":"Phys. Rev. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"012346","DOI":"10.1103\/PhysRevA.94.012346","article-title":"Multipartite entanglement in heterogeneous systems","volume":"94","author":"Goyeneche","year":"2016","journal-title":"Phys. Rev. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"062326","DOI":"10.1103\/PhysRevA.97.062326","article-title":"Entanglement and quantum combinatorial designs","volume":"97","author":"Goyeneche","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"022316","DOI":"10.1103\/PhysRevA.90.022316","article-title":"Genuinely multipartite entangled states and orthogonal arrays","volume":"90","author":"Goyeneche","year":"2014","journal-title":"Phys. Rev. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"975","DOI":"10.1587\/transfun.2021EAP1090","article-title":"k-uniform states and quantum combinatorial designs","volume":"105","author":"Pang","year":"2022","journal-title":"IEICE Trans. Fundam."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1674","DOI":"10.1587\/transfun.2020EAL2007","article-title":"Quantum frequency arrangements, quantum mixed orthogonal arrays and entangled states","volume":"103","author":"Pang","year":"2020","journal-title":"IEICE Trans. Fundam."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"4171","DOI":"10.1109\/TIT.2006.880059","article-title":"Equivalence of decoupling schemes and orthogonal arrays","volume":"52","author":"Wocjan","year":"2006","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"505204","DOI":"10.1088\/1751-8121\/ac3705","article-title":"Quantum combinatorial designs and k-uniform states","volume":"54","author":"Zang","year":"2021","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_23","first-page":"595","article-title":"Orthogonal arrays obtained by orthogonal decomposition of projection matrices","volume":"9","author":"Zhang","year":"1999","journal-title":"Statist. Sin."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/S0012-365X(00)00421-0","article-title":"Orthogonal arrays obtained by the generalized Hadamard product","volume":"238","author":"Zhang","year":"2001","journal-title":"Discrete Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1083","DOI":"10.1007\/s10255-017-0720-z","article-title":"Generalized Latin matrix and construction of orthogonal arrays","volume":"33","author":"Pang","year":"2017","journal-title":"Acta Math. Appl. Sin."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1267","DOI":"10.1587\/transfun.E101.A.1267","article-title":"Construction of asymmetric orthogonal arrays of strength t from orthogonal partition of small orthogonal arrays","volume":"101","author":"Pang","year":"2018","journal-title":"IEICE Trans. Fundam."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"863","DOI":"10.1587\/transfun.E99.A.863","article-title":"The existence of a class of mixed orthogonal arrays","volume":"99","author":"Pang","year":"2016","journal-title":"IEICE Trans. Fundam."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"2870","DOI":"10.1214\/21-AOS2063","article-title":"Construction of mixed orthogonal arrays with high strength","volume":"49","author":"Pang","year":"2021","journal-title":"Ann. Statist."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.spl.2004.03.001","article-title":"Further results on the orthogonal arrays obtained by generalized Hadamard product","volume":"68","author":"Pang","year":"2004","journal-title":"Statist. Probab. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"3895","DOI":"10.1080\/03610926.2019.1591452","article-title":"The Hamming distances of saturated asymmetrical orthogonal arrays with strength 2","volume":"49","author":"Pang","year":"2020","journal-title":"Comm. Statist. Theory Methods"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1186\/s13660-015-0765-6","article-title":"A class of mixed orthogonal arrays obtained from projection matrix inequalities","volume":"2015","author":"Pang","year":"2015","journal-title":"J. Inequal. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1016\/j.jcta.2010.03.013","article-title":"On the existence of orthogonal arrays OA(3,5,4n+2)","volume":"118","author":"Yin","year":"2011","journal-title":"J. Combin. Theory Ser. A"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1002\/jcd.21557","article-title":"Construction of asymmetric orthogonal arrays of strength three via a replacement Method","volume":"25","author":"Zhang","year":"2017","journal-title":"J. Combin. Des."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"200502","DOI":"10.1103\/PhysRevLett.118.200502","article-title":"Absolutely maximally entangled states of seven qubits do not exist","volume":"118","author":"Huber","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"015305","DOI":"10.1088\/1751-8121\/abc9a4","article-title":"Multipartite entanglement states of higher uniformity","volume":"54","author":"Pang","year":"2021","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11128-021-03040-0","article-title":"Quantum k-uniform states for heterogeneous systems from irredundant mixed orthogonal arrays","volume":"20","author":"Pang","year":"2021","journal-title":"Quantum Inf. Process."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1038\/s41534-019-0165-8","article-title":"Two and three-uniform states from irredundant orthogonal arrays","volume":"5","author":"Pang","year":"2019","journal-title":"npj Quantum Inf."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"052330","DOI":"10.1103\/PhysRevA.69.052330","article-title":"Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions","volume":"69","author":"Scott","year":"2004","journal-title":"Phys. Rev. A"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/s11128-020-02978-x","article-title":"New results for 2-uniform states based on irredundant orthogonal arrays","volume":"20","author":"Chen","year":"2021","journal-title":"Quantum Inf. Process."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"032601","DOI":"10.1103\/PhysRevA.104.032601","article-title":"k-uniform quantum information masking","volume":"104","author":"Shi","year":"2021","journal-title":"Phys. Rev. A"},{"key":"ref_41","unstructured":"Edel, Y. (2022, June 01). Some Good Quantum Twisted Code [DB\/OL]. Available online: https:\/\/www.mathi.uni-heidelberg.de\/~yves\/Matritzen\/QTBCH\/QTBCHIndex.html."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Hedayat, A.S., Sloane, N.J.A., and Stufken, J. (1999). Orthogonal Arrays: Theory and Applications, Springer.","DOI":"10.1007\/978-1-4612-1478-6"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1827","DOI":"10.1109\/18.782103","article-title":"Nonbinary quantum codes","volume":"45","author":"Rains","year":"1999","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1360\/012016-23","article-title":"Constructions of mixed orthogonal arrays of strength three (in Chinese)","volume":"47","author":"Chen","year":"2017","journal-title":"Sci. Sin. Math."},{"key":"ref_45","unstructured":"Sloane, N.J.A. (2022, June 01). A Library of Orthogonal Arrays. Available online: http:\/\/neilsloane.com\/oadir\/index.html."},{"key":"ref_46","first-page":"1469","article-title":"Hermitian Self-Orthogonal Constacyclic Codes over F4m","volume":"45","author":"Guan","year":"2017","journal-title":"Acta Electron. Sin."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/7\/1000\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:54:00Z","timestamp":1760140440000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/7\/1000"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,7,19]]},"references-count":46,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2022,7]]}},"alternative-id":["e24071000"],"URL":"https:\/\/doi.org\/10.3390\/e24071000","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,7,19]]}}}