{"title":"论倾斜常环码的等同性和等价性","authors":"Hassan Ou-azzou , Mustapha Najmeddine , Nuh Aydin","doi":"10.1016/j.disc.2024.114279","DOIUrl":null,"url":null,"abstract":"<div><div>In this paper we generalize the notion of <em>n</em>-isometry and <em>n</em>-equivalence relation introduced by Chen et al. in <span><span>[13]</span></span>, <span><span>[12]</span></span> to classify constacyclic codes of length <em>n</em> over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>q</mi></mrow></msub></mrow></msub></math></span>, where <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> is a prime power, to the case of skew constacyclic codes without derivation. We call these relations respectively <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>-equivalence and <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>-isometric relation, where <em>n</em> is the length of the code and <em>σ</em> is an automorphism of the finite field. We compute the number of <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>-equivalence and <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>-isometric classes, and we give conditions on <em>λ</em> and <em>μ</em> for which <span><math><mo>(</mo><mi>σ</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span>-constacyclic codes and <span><math><mo>(</mo><mi>σ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></math></span>-constacyclic codes are equivalent. Under some conditions on <em>n</em> and <em>q</em> we prove that skew constacyclic codes are equivalent to cyclic codes by using properties of our equivalence relation introduced. We also prove that when <em>q</em> is even and <em>σ</em> is the Frobenius automorphism, skew constacyclic codes of length <em>n</em> are equivalent to cyclic codes when <span><math><mi>gcd</mi><mo></mo><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. Finally we give some examples as applications of the theory developed here.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 1","pages":"Article 114279"},"PeriodicalIF":0.7000,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On isometry and equivalence of skew constacyclic codes\",\"authors\":\"Hassan Ou-azzou , Mustapha Najmeddine , Nuh Aydin\",\"doi\":\"10.1016/j.disc.2024.114279\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>In this paper we generalize the notion of <em>n</em>-isometry and <em>n</em>-equivalence relation introduced by Chen et al. in <span><span>[13]</span></span>, <span><span>[12]</span></span> to classify constacyclic codes of length <em>n</em> over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>q</mi></mrow></msub></mrow></msub></math></span>, where <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> is a prime power, to the case of skew constacyclic codes without derivation. We call these relations respectively <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>-equivalence and <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>-isometric relation, where <em>n</em> is the length of the code and <em>σ</em> is an automorphism of the finite field. We compute the number of <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>-equivalence and <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>-isometric classes, and we give conditions on <em>λ</em> and <em>μ</em> for which <span><math><mo>(</mo><mi>σ</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span>-constacyclic codes and <span><math><mo>(</mo><mi>σ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></math></span>-constacyclic codes are equivalent. Under some conditions on <em>n</em> and <em>q</em> we prove that skew constacyclic codes are equivalent to cyclic codes by using properties of our equivalence relation introduced. We also prove that when <em>q</em> is even and <em>σ</em> is the Frobenius automorphism, skew constacyclic codes of length <em>n</em> are equivalent to cyclic codes when <span><math><mi>gcd</mi><mo></mo><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. 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引用次数: 0
摘要
在本文中,我们将 Chen 等人在 [13], [12] 中引入的 n 等式关系和 n 等价关系的概念,用于对有限域 Fq(其中 q=pr 是素幂次)上长度为 n 的共环码进行分类,并将其推广到无派生的倾斜共环码的情况中。我们把这些关系分别称为(n,σ)-等价关系和(n,σ)-等距关系,其中 n 是码的长度,σ 是有限域的自变量。我们计算了(n,σ)等价类和(n,σ)等距类的数量,并给出了λ和μ的条件,在这些条件下,(σ,λ)等价编码和(σ,μ)等价编码是等价的。在 n 和 q 的某些条件下,我们利用所引入的等价关系的性质,证明了倾斜常环码等价于循环码。我们还证明,当 q 为偶数且 σ 为弗罗贝尼斯自变分时,长度为 n 的偏斜自循环码等价于 gcd(n,r)=1 时的循环码。最后,我们将举例说明本文理论的应用。
On isometry and equivalence of skew constacyclic codes
In this paper we generalize the notion of n-isometry and n-equivalence relation introduced by Chen et al. in [13], [12] to classify constacyclic codes of length n over a finite field , where is a prime power, to the case of skew constacyclic codes without derivation. We call these relations respectively -equivalence and -isometric relation, where n is the length of the code and σ is an automorphism of the finite field. We compute the number of -equivalence and -isometric classes, and we give conditions on λ and μ for which -constacyclic codes and -constacyclic codes are equivalent. Under some conditions on n and q we prove that skew constacyclic codes are equivalent to cyclic codes by using properties of our equivalence relation introduced. We also prove that when q is even and σ is the Frobenius automorphism, skew constacyclic codes of length n are equivalent to cyclic codes when . Finally we give some examples as applications of the theory developed here.
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.