(<i>β</i>,<i>γ</i>)-Skew QC Codes with Derivation over a Semi-Local Ring
In this article, we consider a semi-local ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">S</mi><mo>=</mo><msub><mi mathvariant="...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-01-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/1/225 |
| _version_ | 1797436812307726336 |
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| author | Mohammad Ashraf Amal S. Alali Mohd Asim Ghulam Mohammad |
| author_facet | Mohammad Ashraf Amal S. Alali Mohd Asim Ghulam Mohammad |
| author_sort | Mohammad Ashraf |
| collection | DOAJ |
| description | In this article, we consider a semi-local ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">S</mi><mo>=</mo><msub><mi mathvariant="double-struck">F</mi><mi>q</mi></msub><mo>+</mo><mi>u</mi><msub><mi mathvariant="double-struck">F</mi><mi>q</mi></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>=</mo><mi>u</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><msup><mi>p</mi><mi>s</mi></msup></mrow></semantics></math></inline-formula> and <i>p</i> is a prime number. We define a multiplication <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mi>b</mi><mo>=</mo><mi>β</mi><mo>(</mo><mi>b</mi><mo>)</mo><mi>y</mi><mo>+</mo><mi>γ</mi><mo>(</mo><mi>b</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is an automorphism and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> is a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-derivation on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">S</mi></semantics></math></inline-formula> so that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">S</mi><mo>[</mo><mi>y</mi><mo>;</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>]</mo></mrow></semantics></math></inline-formula> becomes a non-commutative ring which is known as skew polynomial ring. We give the characterization of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">S</mi><mo>[</mo><mi>y</mi><mo>;</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>]</mo></mrow></semantics></math></inline-formula> and obtain the most striking results that are better than previous findings. We also determine the structural properties of 1-generator skew cyclic and skew-quasi cyclic codes. Further, We demonstrate remarkable results of the above-mentioned codes over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">S</mi></semantics></math></inline-formula>. Finally, we find the duality of skew cyclic and skew-quasi cyclic codes using a symmetric inner product. These codes are further generalized to double skew cyclic and skew quasi cyclic codes and a table of optimal codes is calculated by MAGMA software. |
| first_indexed | 2024-03-09T11:07:01Z |
| format | Article |
| id | doaj.art-50e1283b7e4448acb6121d669ea95a76 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-09T11:07:01Z |
| publishDate | 2023-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-50e1283b7e4448acb6121d669ea95a762023-12-01T00:53:45ZengMDPI AGSymmetry2073-89942023-01-0115122510.3390/sym15010225(<i>β</i>,<i>γ</i>)-Skew QC Codes with Derivation over a Semi-Local RingMohammad Ashraf0Amal S. Alali1Mohd Asim2Ghulam Mohammad3Department of Mathematics, Aligarh Muslim University, Aligarh 202002, IndiaDepartment of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi ArabiaDepartment of Mathematics, Aligarh Muslim University, Aligarh 202002, IndiaDepartment of Mathematics, Aligarh Muslim University, Aligarh 202002, IndiaIn this article, we consider a semi-local ring <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">S</mi><mo>=</mo><msub><mi mathvariant="double-struck">F</mi><mi>q</mi></msub><mo>+</mo><mi>u</mi><msub><mi mathvariant="double-struck">F</mi><mi>q</mi></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>u</mi><mn>2</mn></msup><mo>=</mo><mi>u</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>=</mo><msup><mi>p</mi><mi>s</mi></msup></mrow></semantics></math></inline-formula> and <i>p</i> is a prime number. We define a multiplication <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mi>b</mi><mo>=</mo><mi>β</mi><mo>(</mo><mi>b</mi><mo>)</mo><mi>y</mi><mo>+</mo><mi>γ</mi><mo>(</mo><mi>b</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> is an automorphism and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>γ</mi></semantics></math></inline-formula> is a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>-derivation on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">S</mi></semantics></math></inline-formula> so that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">S</mi><mo>[</mo><mi>y</mi><mo>;</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>]</mo></mrow></semantics></math></inline-formula> becomes a non-commutative ring which is known as skew polynomial ring. We give the characterization of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="bold">S</mi><mo>[</mo><mi>y</mi><mo>;</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>]</mo></mrow></semantics></math></inline-formula> and obtain the most striking results that are better than previous findings. We also determine the structural properties of 1-generator skew cyclic and skew-quasi cyclic codes. Further, We demonstrate remarkable results of the above-mentioned codes over <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="bold">S</mi></semantics></math></inline-formula>. Finally, we find the duality of skew cyclic and skew-quasi cyclic codes using a symmetric inner product. These codes are further generalized to double skew cyclic and skew quasi cyclic codes and a table of optimal codes is calculated by MAGMA software.https://www.mdpi.com/2073-8994/15/1/225skew polynomial ringskew cyclic codesskew QC codesGray map |
| spellingShingle | Mohammad Ashraf Amal S. Alali Mohd Asim Ghulam Mohammad (<i>β</i>,<i>γ</i>)-Skew QC Codes with Derivation over a Semi-Local Ring Symmetry skew polynomial ring skew cyclic codes skew QC codes Gray map |
| title | (<i>β</i>,<i>γ</i>)-Skew QC Codes with Derivation over a Semi-Local Ring |
| title_full | (<i>β</i>,<i>γ</i>)-Skew QC Codes with Derivation over a Semi-Local Ring |
| title_fullStr | (<i>β</i>,<i>γ</i>)-Skew QC Codes with Derivation over a Semi-Local Ring |
| title_full_unstemmed | (<i>β</i>,<i>γ</i>)-Skew QC Codes with Derivation over a Semi-Local Ring |
| title_short | (<i>β</i>,<i>γ</i>)-Skew QC Codes with Derivation over a Semi-Local Ring |
| title_sort | i β i i γ i skew qc codes with derivation over a semi local ring |
| topic | skew polynomial ring skew cyclic codes skew QC codes Gray map |
| url | https://www.mdpi.com/2073-8994/15/1/225 |
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