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ON ℤpp[u]/k>-CYCLIC CODES AND THEIR WEIGHT ENUMERATORS

  • Received : 2019.07.30
  • Accepted : 2021.02.09
  • Published : 2021.05.01

Abstract

In this paper we study the algebraic structure of ℤpp[u]/k>-cyclic codes, where uk = 0 and p is a prime. A ℤpp[u]/k>-linear code of length (r + s) is an Rk-submodule of ℤrp × Rsk with respect to a suitable scalar multiplication, where Rk = ℤp[u]/k>. Such a code can also be viewed as an Rk-submodule of ℤp[x]/r - 1> × Rk[x]/s - 1>. A new Gray map has been defined on ℤp[u]/k>. We have considered two cases for studying the algebraic structure of ℤpp[u]/k>-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 ℤpp[u]/k>-linear codes. Examples have been given to construct ℤpp[u]/k>-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.

Keywords

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.

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