{"title":"构建 AEAQEC 代码的新方法","authors":"Peng Hu , Xiusheng Liu","doi":"10.1016/j.disc.2024.114202","DOIUrl":null,"url":null,"abstract":"<div><p>Recently, Liu and Liu gave the Singleton bound for pure asymmetric entanglement-assisted quantum error-correcting (AEAQEC) codes. They constructed three new families of AQEAEC codes by means of Vandermonde matrices, generalized Reed-Solomon (GRS) codes and cyclic codes. In this paper, we first exhibit the Singleton bound for any AEAQEC codes. Then we construct AEAQEC codes by two distinct constacyclic codes. By means of repeated-root cyclic codes, we construct new AEAQEC MDS codes. In addition, our methods allow for easily calculating the dimensions, <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>z</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>x</mi></mrow></msub></math></span> and the number <em>c</em> of pre-shared maximally entangled states of AEAQEC codes.</p></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"347 12","pages":"Article 114202"},"PeriodicalIF":0.7000,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"New methods for constructing AEAQEC codes\",\"authors\":\"Peng Hu , Xiusheng Liu\",\"doi\":\"10.1016/j.disc.2024.114202\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Recently, Liu and Liu gave the Singleton bound for pure asymmetric entanglement-assisted quantum error-correcting (AEAQEC) codes. They constructed three new families of AQEAEC codes by means of Vandermonde matrices, generalized Reed-Solomon (GRS) codes and cyclic codes. In this paper, we first exhibit the Singleton bound for any AEAQEC codes. Then we construct AEAQEC codes by two distinct constacyclic codes. By means of repeated-root cyclic codes, we construct new AEAQEC MDS codes. In addition, our methods allow for easily calculating the dimensions, <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>z</mi></mrow></msub></math></span>, <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>x</mi></mrow></msub></math></span> and the number <em>c</em> of pre-shared maximally entangled states of AEAQEC codes.</p></div>\",\"PeriodicalId\":50572,\"journal\":{\"name\":\"Discrete Mathematics\",\"volume\":\"347 12\",\"pages\":\"Article 114202\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-08-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0012365X24003339\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0012365X24003339","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Recently, Liu and Liu gave the Singleton bound for pure asymmetric entanglement-assisted quantum error-correcting (AEAQEC) codes. They constructed three new families of AQEAEC codes by means of Vandermonde matrices, generalized Reed-Solomon (GRS) codes and cyclic codes. In this paper, we first exhibit the Singleton bound for any AEAQEC codes. Then we construct AEAQEC codes by two distinct constacyclic codes. By means of repeated-root cyclic codes, we construct new AEAQEC MDS codes. In addition, our methods allow for easily calculating the dimensions, , and the number c of pre-shared maximally entangled states of AEAQEC codes.
期刊介绍:
Discrete Mathematics provides a common forum for significant research in many areas of discrete mathematics and combinatorics. Among the fields covered by Discrete Mathematics are graph and hypergraph theory, enumeration, coding theory, block designs, the combinatorics of partially ordered sets, extremal set theory, matroid theory, algebraic combinatorics, discrete geometry, matrices, and discrete probability theory.
Items in the journal include research articles (Contributions or Notes, depending on length) and survey/expository articles (Perspectives). Efforts are made to process the submission of Notes (short articles) quickly. The Perspectives section features expository articles accessible to a broad audience that cast new light or present unifying points of view on well-known or insufficiently-known topics.