{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T16:58:16Z","timestamp":1761929896090,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2017,7,18]],"date-time":"2017-07-18T00:00:00Z","timestamp":1500336000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, new construction methods of entanglement-assisted quantum error correction code (EAQECC) from circulant matrices are proposed. We first construct the matrices from two vectors of constraint size, and determine the isotropic subgroup. Then, we also propose a method for calculation of the entanglement subgroup based on standard forms of binary matrices to satisfy the constraint conditions of EAQECC. With isotropic and entanglement subgroups, we determine all the parameters and the minimum distance of the EAQECC. The proposed EAQECC with small lengths are presented to explain the practicality of this construction of EAQECC. Comparison with some earlier constructions of EAQECC shows that the proposed EAQECC is better.<\/jats:p>","DOI":"10.3390\/sym9070122","type":"journal-article","created":{"date-parts":[[2017,7,18]],"date-time":"2017-07-18T10:33:14Z","timestamp":1500373994000},"page":"122","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Minimal-Entanglement Entanglement-Assisted Quantum Error Correction Codes from Modified Circulant Matrices"],"prefix":"10.3390","volume":"9","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4527-3358","authenticated-orcid":false,"given":"Duc Manh","family":"Nguyen","sequence":"first","affiliation":[{"name":"School of Electrical Engineering, University of Ulsan, Ulsan 44610, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1762-5915","authenticated-orcid":false,"given":"Sunghwan","family":"Kim","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, University of Ulsan, Ulsan 44610, Korea"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2017,7,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"305","DOI":"10.3390\/sym3020305","article-title":"Symmetry groups for the decomposition of reversible computers, quantum computers, and computers in between","volume":"3","author":"DeVos","year":"2011","journal-title":"Symmetry"},{"key":"ref_2","unstructured":"Shor, P.W. 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