{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T12:02:20Z","timestamp":1780401740089,"version":"3.54.1"},"reference-count":30,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,4,28]],"date-time":"2024-04-28T00:00:00Z","timestamp":1714262400000},"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":["U21A20428"],"award-info":[{"award-number":["U21A20428"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2023-JC-YB-003"],"award-info":[{"award-number":["2023-JC-YB-003"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["2023-JC-QN-0033"],"award-info":[{"award-number":["2023-JC-QN-0033"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Natural Science Foundation of Shaanxi","award":["U21A20428"],"award-info":[{"award-number":["U21A20428"]}]},{"name":"Natural Science Foundation of Shaanxi","award":["2023-JC-YB-003"],"award-info":[{"award-number":["2023-JC-YB-003"]}]},{"name":"Natural Science Foundation of Shaanxi","award":["2023-JC-QN-0033"],"award-info":[{"award-number":["2023-JC-QN-0033"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Linear complementary dual (LCD) codes, which are a class of linear codes introduced by Massey, have been extensively studied in the literature recently. It has been shown that LCD codes play a role in measures to counter passive and active side-channel analyses on embedded cryptosystems. In this paper, tables are presented of good quaternary Hermitian LCD codes and they are used in the construction of puncturing, shortening and combination codes. The results of this, including three tables of the best-known quaternary Hermitian LCD codes of any length n\u226425 with corresponding dimension k, are presented. In addition, many of these quaternary Hermitian LCD codes given in this paper are optimal and saturate the lower or upper bound of Grassl\u2019s code table, and some of them are nearly optimal.<\/jats:p>","DOI":"10.3390\/e26050373","type":"journal-article","created":{"date-parts":[[2024,4,30]],"date-time":"2024-04-30T04:01:52Z","timestamp":1714449712000},"page":"373","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Optimal Quaternary Hermitian LCD Codes"],"prefix":"10.3390","volume":"26","author":[{"given":"Liangdong","family":"Lu","sequence":"first","affiliation":[{"name":"Fundamentals Department, Air Force Engineering University, Xi\u2019an 710051, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ruihu","family":"Li","sequence":"additional","affiliation":[{"name":"Fundamentals Department, Air Force Engineering University, Xi\u2019an 710051, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yuezhen","family":"Ren","sequence":"additional","affiliation":[{"name":"Fundamentals Department, Air Force Engineering University, Xi\u2019an 710051, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,4,28]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1016\/S0019-9958(64)90438-3","article-title":"Reversible codes","volume":"7","author":"Massey","year":"1964","journal-title":"Inf. Control"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/0012-365X(92)90563-U","article-title":"Linear codes with complementary duals","volume":"106\u2013107","author":"Massey","year":"1992","journal-title":"Discret. Math."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"131","DOI":"10.3934\/amc.2016.10.131","article-title":"Complementary dual codes for counter-measures to side-channel attacks","volume":"10","author":"Carlet","year":"2016","journal-title":"Adv. Math. Commun."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1007\/s11128-014-0830-y","article-title":"Maximal entanglement entanglement-assisted quantum codes constructed from linear codes","volume":"14","author":"Lu","year":"2015","journal-title":"Quantum Inf. Process."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Lv, L., Li, R., Fu, Q., and Li, X. (2015, January 19\u201320). Maximal entanglement entanglement-assisted quantum codes constructed from quaternary BCH codes. Proceedings of the IEEE Advanced Information Technology, Electronic and Automation Control Conference, Chongqing, China.","DOI":"10.1109\/IAEAC.2015.7428647"},{"key":"ref_6","first-page":"4344","article-title":"LCD cyclic codes over finite fields","volume":"63","author":"Ding","year":"2018","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1016\/j.ffa.2016.07.005","article-title":"Quasi-cyclic complementary dual codes","volume":"42","year":"2016","journal-title":"Finite Fields Their Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1016\/0012-365X(94)90283-6","article-title":"The necessary and sufficient condition for a cyclic code to have a complementary dual","volume":"126","author":"Yang","year":"1994","journal-title":"Discret. Math."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"101892","DOI":"10.1016\/j.ffa.2021.101892","article-title":"LCD codes from tridiagonal Toeplitz matrices","volume":"75","author":"Shi","year":"2021","journal-title":"Finite Fields Their Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1016\/j.ffa.2019.04.001","article-title":"Double circulant LCD codes over Z4","volume":"58","author":"Shi","year":"2019","journal-title":"Finite Fields Their Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1016\/j.ffa.2017.11.007","article-title":"Construction of optimal LCD codes over large finite fields","volume":"50","author":"Sok","year":"2018","journal-title":"Finite Fields Their Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"627","DOI":"10.1007\/s12095-021-00545-4","article-title":"LCD and ACD codes over a noncommutative non-unital ring with four elements","volume":"14","author":"Shi","year":"2022","journal-title":"Cryptogr. Commun."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Hou, X., and Oggier, F. (2016, January 10\u201315). On LCD codes and lattices. Proceedings of the IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain.","DOI":"10.1109\/ISIT.2016.7541549"},{"key":"ref_14","first-page":"67","article-title":"On the construction of some LCD codes over finite fields","volume":"9","author":"Lina","year":"2016","journal-title":"Manila J. Sci."},{"key":"ref_15","unstructured":"Zhu, S., Pang, B., and Sun, Z. (2016). The reversible negacyclic codes over finite fields. arXiv."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"719","DOI":"10.1007\/s12095-017-0258-1","article-title":"Some bounds on binary LCD codes","volume":"10","author":"Galvez","year":"2018","journal-title":"Cryptogr. Commun."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"270","DOI":"10.1016\/j.disc.2018.09.034","article-title":"On the classification of linear complementary dual codes","volume":"342","author":"Araya","year":"2019","journal-title":"Discret. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"285","DOI":"10.1007\/s12095-019-00402-5","article-title":"On the minimum weights of binary linear complementary dual codes","volume":"12","author":"Araya","year":"2019","journal-title":"Cryptogr. Commun."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1237","DOI":"10.1142\/S0129054119500242","article-title":"On the Construction of Binary Optimal LCD Codes with Short Length","volume":"30","author":"Fu","year":"2019","journal-title":"Int. J. Found. Comput. Sci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4020","DOI":"10.1109\/TIT.2013.2246274","article-title":"Duality in Entanglement-assisted quantum error correction","volume":"59","author":"Lai","year":"2013","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_21","first-page":"116","article-title":"The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices","volume":"4","author":"Dougherty","year":"2017","journal-title":"Int. J. Inf. Coding Theory"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3010","DOI":"10.1109\/TIT.2018.2789347","article-title":"Linear Codes Over Fq Are Equivalent to LCD Codes for q > 3","volume":"64","author":"Carlet","year":"2018","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"2751","DOI":"10.1109\/TIT.2019.2949040","article-title":"Quaternary Hermitian linear complementary dual codes","volume":"66","author":"Araya","year":"2020","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_24","unstructured":"Grassl, M. (2024, February 28). Bounds on the Minimum Distance of Linear Codes. Available online: http:\/\/www.codetables.de."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"622","DOI":"10.1109\/TIT.2017.2711601","article-title":"Linear Programming Bounds for Entanglement-Assisted Quantum Error-Correcting Codes by Split Weight Enumerators","volume":"64","author":"Lai","year":"2018","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1109\/18.119683","article-title":"Some ternary and quantum codes and associated sphere pachings","volume":"38","author":"Kschischang","year":"1992","journal-title":"IEEE Trans. Inform. Theory"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1016\/j.disc.2003.11.003","article-title":"New bounds for n4(k, d) and classification of some optimal codes over GF(4)","volume":"281","author":"Bouyukliev","year":"2004","journal-title":"Discret. Math."},{"key":"ref_28","unstructured":"Li, R. (2004). Research on Additive Quantum Error Correcting Codes. [Ph.D. Thesis, Northwest Poly-Technical University]."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1750017","DOI":"10.1142\/S0219749917500174","article-title":"Entanglement-assisted quantum codes from quaternary codes of dimension five","volume":"14","author":"Lu","year":"2017","journal-title":"Int. J. Quantum Inf."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1950053","DOI":"10.1142\/S0219749919500539","article-title":"Some optimal entanglement-assisted quantum codes constructed from quaternary Hermitian linear complementary dual codes","volume":"17","author":"Harada","year":"2019","journal-title":"Int. Quantum Inf."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/5\/373\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:35:11Z","timestamp":1760106911000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/5\/373"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,4,28]]},"references-count":30,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2024,5]]}},"alternative-id":["e26050373"],"URL":"https:\/\/doi.org\/10.3390\/e26050373","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,4,28]]}}}