{"title":"有限非链环上循环码的新量子码和LCD码","authors":"Nadeem ur Rehman, Mohd Azmi, Ghulam Mohammad","doi":"10.1016/S0034-4877(23)00027-7","DOIUrl":null,"url":null,"abstract":"<div><p>In this work, we study cyclic codes of length <em>n</em> over a finite commutative non-chain ring\n<span><math><mrow><mi>ℛ</mi><mo>=</mo><msub><mi>F</mi><mi>q</mi></msub><mrow><mo>[</mo><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow><mo>]</mo></mrow><mo>/</mo><mrow><mo>〈</mo><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>−</mo><mi>γ</mi><mi>u</mi><mo>,</mo><msup><mi>v</mi><mn>2</mn></msup><mo>−</mo><mi>ϵ</mi><mi>v</mi><mo>,</mo><mi>u</mi><mi>v</mi><mo>−</mo><mi>v</mi><mi>u</mi></mrow><mo>〉</mo></mrow></mrow></math></span> where\n<span><math><mrow><mi>γ</mi><mo>,</mo><mi>ϵ</mi><mo>∈</mo><msubsup><mi>F</mi><mi>q</mi><mo>*</mo></msubsup></mrow></math></span><span><span> and we find new and better quantum error-correcting codes than previously known quantum error correcting codes. Then certain constraints are imposed on the </span>generator polynomials of cyclic codes, so these codes become linear complementary dual codes (in short LCD codes). We then verify that the Gray image of linear complementary dual codes of length </span><em>n</em> over\n<span><math><mi>ℛ</mi></math></span> is a linear complementary dual code of length 4<em>n</em> over\n<span><math><mrow><msub><mi>F</mi><mi>q</mi></msub></mrow></math></span> by establishing a Gray map.</p></div>","PeriodicalId":49630,"journal":{"name":"Reports on Mathematical Physics","volume":"91 2","pages":"Pages 237-250"},"PeriodicalIF":1.0000,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"NEW QUANTUM AND LCD CODES FROM CYCLIC CODES OVER A FINITE NON-CHAIN RING\",\"authors\":\"Nadeem ur Rehman, Mohd Azmi, Ghulam Mohammad\",\"doi\":\"10.1016/S0034-4877(23)00027-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In this work, we study cyclic codes of length <em>n</em> over a finite commutative non-chain ring\\n<span><math><mrow><mi>ℛ</mi><mo>=</mo><msub><mi>F</mi><mi>q</mi></msub><mrow><mo>[</mo><mrow><mi>u</mi><mo>,</mo><mi>v</mi></mrow><mo>]</mo></mrow><mo>/</mo><mrow><mo>〈</mo><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>−</mo><mi>γ</mi><mi>u</mi><mo>,</mo><msup><mi>v</mi><mn>2</mn></msup><mo>−</mo><mi>ϵ</mi><mi>v</mi><mo>,</mo><mi>u</mi><mi>v</mi><mo>−</mo><mi>v</mi><mi>u</mi></mrow><mo>〉</mo></mrow></mrow></math></span> where\\n<span><math><mrow><mi>γ</mi><mo>,</mo><mi>ϵ</mi><mo>∈</mo><msubsup><mi>F</mi><mi>q</mi><mo>*</mo></msubsup></mrow></math></span><span><span> and we find new and better quantum error-correcting codes than previously known quantum error correcting codes. Then certain constraints are imposed on the </span>generator polynomials of cyclic codes, so these codes become linear complementary dual codes (in short LCD codes). We then verify that the Gray image of linear complementary dual codes of length </span><em>n</em> over\\n<span><math><mi>ℛ</mi></math></span> is a linear complementary dual code of length 4<em>n</em> over\\n<span><math><mrow><msub><mi>F</mi><mi>q</mi></msub></mrow></math></span> by establishing a Gray map.</p></div>\",\"PeriodicalId\":49630,\"journal\":{\"name\":\"Reports on Mathematical Physics\",\"volume\":\"91 2\",\"pages\":\"Pages 237-250\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2023-04-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Reports on Mathematical Physics\",\"FirstCategoryId\":\"101\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0034487723000277\",\"RegionNum\":4,\"RegionCategory\":\"物理与天体物理\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Reports on Mathematical Physics","FirstCategoryId":"101","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0034487723000277","RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
NEW QUANTUM AND LCD CODES FROM CYCLIC CODES OVER A FINITE NON-CHAIN RING
In this work, we study cyclic codes of length n over a finite commutative non-chain ring
where
and we find new and better quantum error-correcting codes than previously known quantum error correcting codes. Then certain constraints are imposed on the generator polynomials of cyclic codes, so these codes become linear complementary dual codes (in short LCD codes). We then verify that the Gray image of linear complementary dual codes of length n over
is a linear complementary dual code of length 4n over
by establishing a Gray map.
期刊介绍：
Reports on Mathematical Physics publish papers in theoretical physics which present a rigorous mathematical approach to problems of quantum and classical mechanics and field theories, relativity and gravitation, statistical physics, thermodynamics, mathematical foundations of physical theories, etc. Preferred are papers using modern methods of functional analysis, probability theory, differential geometry, algebra and mathematical logic. Papers without direct connection with physics will not be accepted. Manuscripts should be concise, but possibly complete in presentation and discussion, to be comprehensible not only for mathematicians, but also for mathematically oriented theoretical physicists. All papers should describe original work and be written in English.
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