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Title: On ℤpℤp[u]/〈uk〉-cyclic codes and their weight enumerators
Authors: Bhaintwal, Maheshanand
Biswas S.
Published in: Journal of the Korean Mathematical Society
Abstract: In this paper we study the algebraic structure of ℤpℤp[u]/ 〈uk 〉-cyclic codes, where uk = 0 and p is a prime. A ℤpℤp[u]/〈uk 〉-linear code of length (r + s) is an Rk-submodule of ℤrp× Rsk with respect to a suitable scalar multiplication, where Rk = ℤp[u]/〈uk 〉. Such a code can also be viewed as an Rk-submodule of ℤp[x]/〈xr − 1〉 × Rk [x]/〈xs − 1〉. A new Gray map has been defined on ℤp[u]/〈uk 〉. We have considered two cases for studying the algebraic structure of ℤpℤp[u]/〈uk 〉-cyclic codes, and determined the generator polynomials and minimal spanning sets of these codes in both the cases. In the first case, we have considered (r, p) = 1 and (s, p) ≠ 1, and in the second case we consider (r, p) = 1 and (s, p) = 1. We have established the MacWilliams identity for complete weight enumerators of ℤpℤp[u]/〈uk 〉-linear codes. Examples have been given to construct ℤpℤp[u]/〈uk 〉-cyclic codes, through which we get codes over ℤp using the Gray map. Some optimal p-ary codes have been obtained in this way. An example has also been given to illustrate the use of MacWilliams identity. © 2021 Korean Mathematical Society.
Citation: Journal of the Korean Mathematical Society, 58(3): 571-595
Issue Date: 2021
Publisher: Korean Mathematical Society
Keywords: Gray map
Weight enumerators
ℤpℤp[u]/〈uk 〉-cyclic codes
ℤpℤp[u]/〈uk 〉-linear codes
ISSN: 3049914
Author Scopus IDs: 32867546000
Author Affiliations: Bhaintwal, M., Department of Mathematics, Indian Institute of Technology, Roorkee Roorkee, 247667, India
Biswas, S., Department of Mathematics, Indian Institute of Technology, Roorkee Roorkee, 247667, India
Funding Details: Acknowledgement. The authors are thankful to the anonymous referee for his/her careful reading of the manuscript and helpful suggestions that greatly improved the final presentation of the manuscript. The second author would like to thank Ministry of Human Resource Development (MHRD), India, for providing financial support. Ministry of Education, India, MoE
Appears in Collections:Journal Publications [MA]

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