{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T02:54:30Z","timestamp":1747191270232,"version":"3.40.5"},"reference-count":26,"publisher":"Wiley","license":[{"start":{"date-parts":[[2023,3,22]],"date-time":"2023-03-22T00:00:00Z","timestamp":1679443200000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Zhengzhou Special Fund for Basic Research and Applied Basic Research","award":["ZZSZX202111","212102210573"],"award-info":[{"award-number":["ZZSZX202111","212102210573"]}]},{"name":"Key Technologies Research and Development Program of Henan Province","award":["ZZSZX202111","212102210573"],"award-info":[{"award-number":["ZZSZX202111","212102210573"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Quantum Engineering"],"published-print":{"date-parts":[[2023,3,22]]},"abstract":"<jats:p>Let <jats:inline-formula>\n                     <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\">\n                        <a:msub>\n                           <a:mrow>\n                              <a:mi mathvariant=\"fraktur\">R<\/a:mi>\n                           <\/a:mrow>\n                           <a:mrow>\n                              <a:mi>k<\/a:mi>\n                           <\/a:mrow>\n                        <\/a:msub>\n                        <a:mo>=<\/a:mo>\n                        <a:msub>\n                           <a:mrow>\n                              <a:mi mathvariant=\"double-struck\">F<\/a:mi>\n                           <\/a:mrow>\n                           <a:mrow>\n                              <a:mi>q<\/a:mi>\n                           <\/a:mrow>\n                        <\/a:msub>\n                        <a:mfenced open=\"[\" close=\"]\" separators=\"|\">\n                           <a:mrow>\n                              <a:msub>\n                                 <a:mrow>\n                                    <a:mi>u<\/a:mi>\n                                 <\/a:mrow>\n                                 <a:mrow>\n                                    <a:mn>1<\/a:mn>\n                                 <\/a:mrow>\n                              <\/a:msub>\n                              <a:mo>,<\/a:mo>\n                              <a:msub>\n                                 <a:mrow>\n                                    <a:mi>u<\/a:mi>\n                                 <\/a:mrow>\n                                 <a:mrow>\n                                    <a:mn>2<\/a:mn>\n                                 <\/a:mrow>\n                              <\/a:msub>\n                              <a:mo>,<\/a:mo>\n                              <a:mo>\u22ef<\/a:mo>\n                              <a:mo>,<\/a:mo>\n                              <a:msub>\n                                 <a:mrow>\n                                    <a:mi>u<\/a:mi>\n                                 <\/a:mrow>\n                                 <a:mrow>\n                                    <a:mi>k<\/a:mi>\n                                 <\/a:mrow>\n                              <\/a:msub>\n                           <\/a:mrow>\n                        <\/a:mfenced>\n                        <a:mo>\/<\/a:mo>\n                        <a:mo>&lt;<\/a:mo>\n                        <a:msubsup>\n                           <a:mrow>\n                              <a:mi>u<\/a:mi>\n                           <\/a:mrow>\n                           <a:mrow>\n                              <a:mi>i<\/a:mi>\n                           <\/a:mrow>\n                           <a:mrow>\n                              <a:mn>2<\/a:mn>\n                           <\/a:mrow>\n                        <\/a:msubsup>\n                        <a:mo>=<\/a:mo>\n                        <a:msub>\n                           <a:mrow>\n                              <a:mi>\u03b1<\/a:mi>\n                           <\/a:mrow>\n                           <a:mrow>\n                              <a:mi>i<\/a:mi>\n                           <\/a:mrow>\n                        <\/a:msub>\n                        <a:msub>\n                           <a:mrow>\n                              <a:mi>u<\/a:mi>\n                           <\/a:mrow>\n                           <a:mrow>\n                              <a:mi>i<\/a:mi>\n                           <\/a:mrow>\n                        <\/a:msub>\n                        <a:mo>,<\/a:mo>\n                        <a:msub>\n                           <a:mrow>\n                              <a:mi>u<\/a:mi>\n                           <\/a:mrow>\n                           <a:mrow>\n                              <a:mi>i<\/a:mi>\n                           <\/a:mrow>\n                        <\/a:msub>\n                        <a:msub>\n                           <a:mrow>\n                              <a:mi>u<\/a:mi>\n                           <\/a:mrow>\n                           <a:mrow>\n                              <a:mi>j<\/a:mi>\n                           <\/a:mrow>\n                        <\/a:msub>\n                        <a:mo>=<\/a:mo>\n                        <a:msub>\n                           <a:mrow>\n                              <a:mi>u<\/a:mi>\n                           <\/a:mrow>\n                           <a:mrow>\n                              <a:mi>j<\/a:mi>\n                           <\/a:mrow>\n                        <\/a:msub>\n                        <a:msub>\n                           <a:mrow>\n                              <a:mi>u<\/a:mi>\n                           <\/a:mrow>\n                           <a:mrow>\n                              <a:mi>i<\/a:mi>\n                           <\/a:mrow>\n                        <\/a:msub>\n                        <a:mo>=<\/a:mo>\n                        <a:mn>0<\/a:mn>\n                        <a:mo>&gt;<\/a:mo>\n                     <\/a:math>\n                  <\/jats:inline-formula>, where <jats:inline-formula>\n                     <h:math xmlns:h=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\">\n                        <h:mi>q<\/h:mi>\n                        <h:mo>=<\/h:mo>\n                        <h:msup>\n                           <h:mrow>\n                              <h:mi>p<\/h:mi>\n                           <\/h:mrow>\n                           <h:mrow>\n                              <h:mi>m<\/h:mi>\n                           <\/h:mrow>\n                        <\/h:msup>\n                     <\/h:math>\n                  <\/jats:inline-formula>, <jats:inline-formula>\n                     <j:math xmlns:j=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\">\n                        <j:mi>p<\/j:mi>\n                     <\/j:math>\n                  <\/jats:inline-formula> is an odd prime, <jats:inline-formula>\n                     <l:math xmlns:l=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\">\n                        <l:msub>\n                           <l:mrow>\n                              <l:mi>\u03b1<\/l:mi>\n                           <\/l:mrow>\n                           <l:mrow>\n                              <l:mi>i<\/l:mi>\n                           <\/l:mrow>\n                        <\/l:msub>\n                     <\/l:math>\n                  <\/jats:inline-formula> is a unit over <jats:inline-formula>\n                     <n:math xmlns:n=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\">\n                        <n:msub>\n                           <n:mrow>\n                              <n:mi mathvariant=\"double-struck\">F<\/n:mi>\n                           <\/n:mrow>\n                           <n:mrow>\n                              <n:mi>q<\/n:mi>\n                           <\/n:mrow>\n                        <\/n:msub>\n                     <\/n:math>\n                  <\/jats:inline-formula>, and <jats:inline-formula>\n                     <q:math xmlns:q=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\">\n                        <q:mi>i<\/q:mi>\n                        <q:mo>,<\/q:mo>\n                        <q:mi>j<\/q:mi>\n                        <q:mo>=<\/q:mo>\n                        <q:mn>1,2<\/q:mn>\n                        <q:mo>,<\/q:mo>\n                        <q:mo>\u22ef<\/q:mo>\n                        <q:mo>,<\/q:mo>\n                        <q:mi>k<\/q:mi>\n                        <q:mtext>.<\/q:mtext>\n                     <\/q:math>\n                  <\/jats:inline-formula> In this article, we define a Gray map from <jats:inline-formula>\n                     <s:math xmlns:s=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M10\">\n                        <s:msubsup>\n                           <s:mrow>\n                              <s:mi mathvariant=\"fraktur\">R<\/s:mi>\n                           <\/s:mrow>\n                           <s:mrow>\n                              <s:mi>k<\/s:mi>\n                           <\/s:mrow>\n                           <s:mrow>\n                              <s:mi>n<\/s:mi>\n                           <\/s:mrow>\n                        <\/s:msubsup>\n                     <\/s:math>\n                  <\/jats:inline-formula> to <jats:inline-formula>\n                     <v:math xmlns:v=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M11\">\n                        <v:msubsup>\n                           <v:mrow>\n                              <v:mi mathvariant=\"double-struck\">F<\/v:mi>\n                           <\/v:mrow>\n                           <v:mrow>\n                              <v:mi>q<\/v:mi>\n                           <\/v:mrow>\n                           <v:mrow>\n                              <v:mfenced open=\"(\" close=\")\" separators=\"|\">\n                                 <v:mrow>\n                                    <v:mi>k<\/v:mi>\n                                    <v:mo>+<\/v:mo>\n                                    <v:mn>1<\/v:mn>\n                                 <\/v:mrow>\n                              <\/v:mfenced>\n                              <v:mi>n<\/v:mi>\n                           <\/v:mrow>\n                        <\/v:msubsup>\n                     <\/v:math>\n                  <\/jats:inline-formula>, we study the structure of skew <jats:inline-formula>\n                     <bb:math xmlns:bb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M12\">\n                        <bb:mi>\u03f1<\/bb:mi>\n                     <\/bb:math>\n                  <\/jats:inline-formula>-<jats:inline-formula>\n                     <db:math xmlns:db=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M13\">\n                        <db:mi>\u03bb<\/db:mi>\n                     <\/db:math>\n                  <\/jats:inline-formula>-constacyclic codes over <jats:inline-formula>\n                     <fb:math xmlns:fb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M14\">\n                        <fb:msub>\n                           <fb:mrow>\n                              <fb:mi mathvariant=\"fraktur\">R<\/fb:mi>\n                           <\/fb:mrow>\n                           <fb:mrow>\n                              <fb:mi>k<\/fb:mi>\n                           <\/fb:mrow>\n                        <\/fb:msub>\n                     <\/fb:math>\n                  <\/jats:inline-formula>, and then we give the necessary and sufficient conditions for skew <jats:inline-formula>\n                     <ib:math xmlns:ib=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M15\">\n                        <ib:mi>\u03f1<\/ib:mi>\n                     <\/ib:math>\n                  <\/jats:inline-formula>-<jats:inline-formula>\n                     <kb:math xmlns:kb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M16\">\n                        <kb:mi>\u03bb<\/kb:mi>\n                     <\/kb:math>\n                  <\/jats:inline-formula>-constacyclic codes over <jats:inline-formula>\n                     <mb:math xmlns:mb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M17\">\n                        <mb:msub>\n                           <mb:mrow>\n                              <mb:mi mathvariant=\"fraktur\">R<\/mb:mi>\n                           <\/mb:mrow>\n                           <mb:mrow>\n                              <mb:mi>k<\/mb:mi>\n                           <\/mb:mrow>\n                        <\/mb:msub>\n                     <\/mb:math>\n                  <\/jats:inline-formula> to satisfy dual containing. Further, we have obtained some new nonbinary quantum codes from skew <jats:inline-formula>\n                     <pb:math xmlns:pb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M18\">\n                        <pb:mi>\u03f1<\/pb:mi>\n                     <\/pb:math>\n                  <\/jats:inline-formula>-<jats:inline-formula>\n                     <rb:math xmlns:rb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M19\">\n                        <rb:mi>\u03bb<\/rb:mi>\n                     <\/rb:math>\n                  <\/jats:inline-formula>-constacyclic over <jats:inline-formula>\n                     <tb:math xmlns:tb=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M20\">\n                        <tb:msub>\n                           <tb:mrow>\n                              <tb:mi mathvariant=\"fraktur\">R<\/tb:mi>\n                           <\/tb:mrow>\n                           <tb:mrow>\n                              <tb:mi>k<\/tb:mi>\n                           <\/tb:mrow>\n                        <\/tb:msub>\n                     <\/tb:math>\n                  <\/jats:inline-formula> by using the CSS construction.<\/jats:p>","DOI":"10.1155\/2023\/8196228","type":"journal-article","created":{"date-parts":[[2023,3,23]],"date-time":"2023-03-23T09:58:54Z","timestamp":1679565534000},"page":"1-7","source":"Crossref","is-referenced-by-count":0,"title":["Quantum Codes Obtained from Skew <a:math xmlns:a=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\">\n                     <a:mi>\u03f1<\/a:mi>\n                  <\/a:math>-<c:math xmlns:c=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\">\n                     <c:mi>\u03bb<\/c:mi>\n                  <\/c:math>-Constacyclic Codes over <e:math xmlns:e=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\">\n                     <e:msub>\n                        <e:mrow>\n                           <e:mi mathvariant=\"fraktur\">R<\/e:mi>\n                        <\/e:mrow>\n                        <e:mrow>\n                           <e:mi>k<\/e:mi>\n                        <\/e:mrow>\n                     <\/e:msub>\n                  <\/e:math>"],"prefix":"10.1155","volume":"2023","author":[{"given":"Bo","family":"Kong","sequence":"first","affiliation":[{"name":"School of Statistics and Mathematics, Henan Finance University, Zhengzhou 450046, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4652-3555","authenticated-orcid":true,"given":"Xiying","family":"Zheng","sequence":"additional","affiliation":[{"name":"Faculty of Engineering, Huanghe Science and Technology College, Zhengzhou 450063, China"}]},{"given":"Jie","family":"Liu","sequence":"additional","affiliation":[{"name":"Faculty of Engineering, Huanghe Science and Technology College, Zhengzhou 450063, China"}]},{"given":"Hongjing","family":"Jiang","sequence":"additional","affiliation":[{"name":"Faculty of Engineering, Huanghe Science and Technology College, Zhengzhou 450063, China"}]}],"member":"311","reference":[{"doi-asserted-by":"publisher","key":"1","DOI":"10.1103\/PhysRevA.54.1098"},{"doi-asserted-by":"publisher","key":"2","DOI":"10.1103\/PhysRevA.54.4741"},{"doi-asserted-by":"publisher","key":"3","DOI":"10.1103\/PhysRevLett.77.793"},{"doi-asserted-by":"publisher","key":"4","DOI":"10.1109\/18.681315"},{"doi-asserted-by":"publisher","key":"5","DOI":"10.1142\/S0219749904000079"},{"doi-asserted-by":"publisher","key":"6","DOI":"10.1109\/TIT.2006.883612"},{"doi-asserted-by":"publisher","key":"7","DOI":"10.1142\/S1793830918500465"},{"doi-asserted-by":"publisher","key":"8","DOI":"10.1007\/s11128-021-03052-w"},{"doi-asserted-by":"publisher","key":"9","DOI":"10.1007\/s11786-019-00426-3"},{"doi-asserted-by":"publisher","key":"10","DOI":"10.1007\/s12190-020-01407-7"},{"doi-asserted-by":"publisher","key":"11","DOI":"10.1007\/s00200-018-0366-y"},{"doi-asserted-by":"publisher","key":"12","DOI":"10.1142\/S0219749915500318"},{"doi-asserted-by":"publisher","key":"13","DOI":"10.1007\/s12190-018-01230-1"},{"doi-asserted-by":"publisher","key":"14","DOI":"10.1515\/math-2022-0459"},{"doi-asserted-by":"publisher","key":"15","DOI":"10.1007\/s00200-007-0043-z"},{"doi-asserted-by":"publisher","key":"16","DOI":"10.1016\/j.jsc.2007.11.008"},{"doi-asserted-by":"publisher","key":"17","DOI":"10.1504\/IJICOT.2011.044674"},{"doi-asserted-by":"publisher","key":"18","DOI":"10.3934\/amc.2012.6.39"},{"doi-asserted-by":"publisher","key":"19","DOI":"10.1155\/2016\/3965789"},{"doi-asserted-by":"publisher","key":"20","DOI":"10.1007\/s11128-019-2384-5"},{"doi-asserted-by":"publisher","key":"21","DOI":"10.1016\/j.disc.2019.111737"},{"doi-asserted-by":"publisher","key":"22","DOI":"10.1007\/s11128-020-02700-x"},{"doi-asserted-by":"publisher","key":"23","DOI":"10.1140\/epjp\/s13360-022-02429-9"},{"doi-asserted-by":"publisher","key":"24","DOI":"10.1007\/s10773-021-04910-0"},{"doi-asserted-by":"publisher","key":"25","DOI":"10.1016\/j.disc.2020.112189"},{"doi-asserted-by":"publisher","key":"26","DOI":"10.1007\/s12095-022-00594-3"}],"container-title":["Quantum Engineering"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/que\/2023\/8196228.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/que\/2023\/8196228.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/que\/2023\/8196228.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,23]],"date-time":"2023-03-23T09:59:01Z","timestamp":1679565541000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.hindawi.com\/journals\/que\/2023\/8196228\/"}},"subtitle":[],"editor":[{"given":"Liuguo","family":"Yin","sequence":"additional","affiliation":[]}],"short-title":[],"issued":{"date-parts":[[2023,3,22]]},"references-count":26,"alternative-id":["8196228","8196228"],"URL":"https:\/\/doi.org\/10.1155\/2023\/8196228","relation":{},"ISSN":["2577-0470"],"issn-type":[{"type":"electronic","value":"2577-0470"}],"subject":[],"published":{"date-parts":[[2023,3,22]]}}}