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;

Let F<sub>2(m)</sub> be a finite field of 2<sup>m</sup> elements, &#x03BB; and k be integers satisfying &#x03BB;, k &#x2265; 2 and denote R = F<sub>2(m)</sub>[u]/(u<sup>2&#x03BB;</sup>). Let &#x03B4;, &#x03B1; &#x2208; F<...

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Main Authors: Yuan Cao, Yonglin Cao, Hai Q. Dinh, Tushar Bag, Woraphon Yamaka
Format: Article
Language:English
Published: IEEE 2020-01-01
Series:IEEE Access
Subjects:
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, &#x03BB; and k be integers satisfying &#x03BB;, k &#x2265; 2 and denote R = F<sub>2(m)</sub>[u]/(u<sup>2&#x03BB;</sup>). Let &#x03B4;, &#x03B1; &#x2208; F<sub>2(m)</sub><sup>&#x00D7;</sup>. For any odd positive integer n, we give an explicit representation and enumeration for all distinct (&#x03B4; +&#x03B1;u<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 (&#x03B4; + &#x03B1;u<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]/&#x2329;x<sup>2(k)n</sup> - (&#x03B4; + &#x03B1;u<sup>2</sup>)&#x232A;. As an example, we listed all 135 distinct (1 + u<sup>2</sup>)-constacyclic codes of length 4 over F<sub>2</sub>[u]/&#x2329;u<sup>4</sup>&#x232A;, and apply our results to determine all 11 self-dual codes among them.
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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 (&#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;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, &#x03BB; and k be integers satisfying &#x03BB;, k &#x2265; 2 and denote R = F<sub>2(m)</sub>[u]/(u<sup>2&#x03BB;</sup>). Let &#x03B4;, &#x03B1; &#x2208; F<sub>2(m)</sub><sup>&#x00D7;</sup>. For any odd positive integer n, we give an explicit representation and enumeration for all distinct (&#x03B4; +&#x03B1;u<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 (&#x03B4; + &#x03B1;u<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]/&#x2329;x<sup>2(k)n</sup> - (&#x03B4; + &#x03B1;u<sup>2</sup>)&#x232A;. As an example, we listed all 135 distinct (1 + u<sup>2</sup>)-constacyclic codes of length 4 over F<sub>2</sub>[u]/&#x2329;u<sup>4</sup>&#x232A;, 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 (&#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;
IEEE Access
Type 2 constacyclic code
linear code
repeated-root code
finite chain ring
title 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;
title_full 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;
title_fullStr 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;
title_full_unstemmed 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;
title_short 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;
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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AT yonglincao explicitrepresentationandenumerationofrepeatedrootx03b4x002bitalicx03b1uitalicx00b2constacycliccodesoverfx2082supitalicmitalicsupitalicuitalicx2039italicuitalicsup2x03bbsupx203a
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