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http://dx.doi.org/10.4134/JKMS.j190536

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

Bhaintwal, Maheshanand (Department of Mathematics Indian Institute of Technology Roorkee)
Biswas, Soumak (Department of Mathematics Indian Institute of Technology Roorkee)
Publication Information
Journal of the Korean Mathematical Society / v.58, no.3, 2021 , pp. 571-595 More about this Journal
Abstract
In this paper we study the algebraic structure of ℤpℤp[u]/k>-cyclic codes, where uk = 0 and p is a prime. A ℤpℤp[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 ℤpℤp[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 ℤpℤp[u]/k>-linear codes. Examples have been given to construct ℤpℤp[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
${\mathbb{Z}}_p{\mathbb{Z}}_p[u]$/<$u^k$>-linear codes; ${\mathbb{Z}}_p{\mathbb{Z}}_p[u]$/<$u^k$>-cyclic codes; Gray map; weight enumerators;
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