Explicit Representation and Enumeration of Repeated-Root (δ + <italic>αu</italic>²)-Constacyclic Codes Over F₂<sup><italic>m</italic></sup>[<italic>u</italic>]/‹<italic>u</italic><sup>2λ</sup>›
Let F<sub>2(m)</sub> be a finite field of 2<sup>m</sup> elements, λ and k be integers satisfying λ, k ≥ 2 and denote R = F<sub>2(m)</sub>[u]/(u<sup>2λ</sup>). Let δ, α ∈ F<...
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IEEE
2020-01-01
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Online Access: | https://ieeexplore.ieee.org/document/9039567/ |
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author | Yuan Cao Yonglin Cao Hai Q. Dinh Tushar Bag Woraphon Yamaka |
author_facet | Yuan Cao Yonglin Cao Hai Q. Dinh Tushar Bag Woraphon Yamaka |
author_sort | Yuan Cao |
collection | DOAJ |
description | Let F<sub>2(m)</sub> be a finite field of 2<sup>m</sup> elements, λ and k be integers satisfying λ, k ≥ 2 and denote R = F<sub>2(m)</sub>[u]/(u<sup>2λ</sup>). Let δ, α ∈ F<sub>2(m)</sub><sup>×</sup>. For any odd positive integer n, we give an explicit representation and enumeration for all distinct (δ +αu<sup>2</sup>)-constacyclic codes over R of length 2<sup>k</sup>n, and provide a clear formula to count the number of all these codes. In particular, we conclude that every (δ + αu<sup>2</sup>)-constacyclic code over R of length 2<sup>k</sup>n is an ideal generated by at most 2 polynomials in the ring R[x]/〈x<sup>2(k)n</sup> - (δ + αu<sup>2</sup>)〉. As an example, we listed all 135 distinct (1 + u<sup>2</sup>)-constacyclic codes of length 4 over F<sub>2</sub>[u]/〈u<sup>4</sup>〉, and apply our results to determine all 11 self-dual codes among them. |
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institution | Directory Open Access Journal |
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language | English |
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spelling | doaj.art-e52f280e83e847d28bc4300d6a7e75b42022-12-21T22:01:17ZengIEEEIEEE Access2169-35362020-01-018555505556210.1109/ACCESS.2020.29814539039567Explicit Representation and Enumeration of Repeated-Root (δ + <italic>αu</italic>²)-Constacyclic Codes Over F₂<sup><italic>m</italic></sup>[<italic>u</italic>]/‹<italic>u</italic><sup>2λ</sup>›Yuan Cao0https://orcid.org/0000-0002-3089-0046Yonglin Cao1https://orcid.org/0000-0002-3682-6483Hai Q. Dinh2https://orcid.org/0000-0002-6487-8803Tushar Bag3https://orcid.org/0000-0002-7613-8351Woraphon Yamaka4https://orcid.org/0000-0002-0787-1437School of Mathematics and Statistics, Shandong University of Technology, Zibo, ChinaSchool of Mathematics and Statistics, Shandong University of Technology, Zibo, ChinaDivision of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, VietnamDepartment of Mathematics, Indian Institute of Technology Patna, Patna, IndiaCentre of Excellence in Econometrics, Chiang Mai University, Chiang Mai, ThailandLet F<sub>2(m)</sub> be a finite field of 2<sup>m</sup> elements, λ and k be integers satisfying λ, k ≥ 2 and denote R = F<sub>2(m)</sub>[u]/(u<sup>2λ</sup>). Let δ, α ∈ F<sub>2(m)</sub><sup>×</sup>. For any odd positive integer n, we give an explicit representation and enumeration for all distinct (δ +αu<sup>2</sup>)-constacyclic codes over R of length 2<sup>k</sup>n, and provide a clear formula to count the number of all these codes. In particular, we conclude that every (δ + αu<sup>2</sup>)-constacyclic code over R of length 2<sup>k</sup>n is an ideal generated by at most 2 polynomials in the ring R[x]/〈x<sup>2(k)n</sup> - (δ + αu<sup>2</sup>)〉. As an example, we listed all 135 distinct (1 + u<sup>2</sup>)-constacyclic codes of length 4 over F<sub>2</sub>[u]/〈u<sup>4</sup>〉, and apply our results to determine all 11 self-dual codes among them.https://ieeexplore.ieee.org/document/9039567/Type 2 constacyclic codelinear coderepeated-root codefinite chain ring |
spellingShingle | Yuan Cao Yonglin Cao Hai Q. Dinh Tushar Bag Woraphon Yamaka Explicit Representation and Enumeration of Repeated-Root (δ + <italic>αu</italic>²)-Constacyclic Codes Over F₂<sup><italic>m</italic></sup>[<italic>u</italic>]/‹<italic>u</italic><sup>2λ</sup>› IEEE Access Type 2 constacyclic code linear code repeated-root code finite chain ring |
title | Explicit Representation and Enumeration of Repeated-Root (δ + <italic>αu</italic>²)-Constacyclic Codes Over F₂<sup><italic>m</italic></sup>[<italic>u</italic>]/‹<italic>u</italic><sup>2λ</sup>› |
title_full | Explicit Representation and Enumeration of Repeated-Root (δ + <italic>αu</italic>²)-Constacyclic Codes Over F₂<sup><italic>m</italic></sup>[<italic>u</italic>]/‹<italic>u</italic><sup>2λ</sup>› |
title_fullStr | Explicit Representation and Enumeration of Repeated-Root (δ + <italic>αu</italic>²)-Constacyclic Codes Over F₂<sup><italic>m</italic></sup>[<italic>u</italic>]/‹<italic>u</italic><sup>2λ</sup>› |
title_full_unstemmed | Explicit Representation and Enumeration of Repeated-Root (δ + <italic>αu</italic>²)-Constacyclic Codes Over F₂<sup><italic>m</italic></sup>[<italic>u</italic>]/‹<italic>u</italic><sup>2λ</sup>› |
title_short | Explicit Representation and Enumeration of Repeated-Root (δ + <italic>αu</italic>²)-Constacyclic Codes Over F₂<sup><italic>m</italic></sup>[<italic>u</italic>]/‹<italic>u</italic><sup>2λ</sup>› |
title_sort | explicit representation and enumeration of repeated root x03b4 x002b italic x03b1 u italic x00b2 constacyclic codes over f x2082 sup italic m italic sup italic u italic x2039 italic u italic sup 2 x03bb sup x203a |
topic | Type 2 constacyclic code linear code repeated-root code finite chain ring |
url | https://ieeexplore.ieee.org/document/9039567/ |
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