<abstract xmlns="http://www.w3.org/1999/xhtml">

Let <bold>S</bold> = ℤ<italic>p</italic>[<italic>u, v</italic>]<italic>/〈u</italic>2, <italic>v</italic>2, <italic>uv − uv〉</italic> be a semi-local ring, where <italic>p</italic> is a prime number. In the present article, we determine the generating sets of <bold>S</bold> and use them to construct the structures of ℤ<italic>p</italic><bold>S</bold>-additive cyclic and constacyclic codes. The minimal polynomials and spanning sets of ℤ<italic>p</italic><bold>S</bold>-additive cyclic and constacyclic codes are also determined. These codes are identified as <bold>S</bold>[<italic>y</italic>]-submodules of the ring <bold>S</bold>β1, β2 = ℤ<italic>p</italic>[<italic>y</italic>]<italic>/〈y</italic><italic>β</italic>1 <italic>−</italic> 1<italic>〉 ×</italic> <bold>S</bold>[<italic>y</italic>]<italic>/〈y</italic><italic>β</italic>2 <italic>−</italic> 1<italic>〉</italic>. Some results that represent the relationship between the minimal polynomials of ℤ<italic>p</italic><bold>S</bold>-additive cyclic codes and their duals have been obtained. Furthermore, optimal ℤ<italic>p</italic><bold>S</bold>-additive codes and maximum distance separable codes have been evaluated (see Table 1). Finally, we use MAGMA software to find the parameters of Optimal and MDSS codes.
</abstract>

Let S = ℤp[u, v]/〈u2, v2, uv − uv〉 be a semi-local ring, where p is a prime number. In the present article, we determine the generating sets of S and use them to construct the structures of ℤpS-additive cyclic and constacyclic codes. The minimal polynomials and spanning sets of ℤpS-additive cyclic and constacyclic codes are also determined. These codes are identified as S[y]-submodules of the ring Sβ1, β2 = ℤp[y]/〈yβ1 − 1〉 × S[y]/〈yβ2 − 1〉. Some results that represent the relationship between the minimal polynomials of ℤpS-additive cyclic codes and their duals have been obtained. Furthermore, optimal ℤpS-additive codes and maximum distance separable codes have been evaluated (see Table 1). Finally, we use MAGMA software to find the parameters of Optimal and MDSS codes.

The study of ℤ<italic>p</italic>ℤ<italic>p</italic>[<italic>u, v</italic>]-additive cyclic codes and their application in obtaining Optimal and MDSS codes

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

The study of ℤℤ[]-additive cyclic codes and their application in obtaining Optimal and MDSS codes

Let S = ℤp[u, v]/〈u2, v2, uv − uv〉 be a semi-local ring, where p is a prime number. In the present article, we determine the generating sets of S and use them...

The study of ℤ_pℤ_p[u, v]-additive cyclic codes and their application in obtaining Optimal and MDSS codes

Published Online: May 27, 2024

Page range: 19 - 44

Received: Jan 11, 2023

Accepted: Apr 18, 2023

DOI: https://doi.org/10.2478/auom-2024-0002

Keywords
polynomial ring, additive cyclic codes, Gray map

© 2024 Mohammad Ashraf et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

The study of ℤpℤp[u, v]-additive cyclic codes and their application in obtaining Optimal and MDSS codes

Published Online: May 27, 2024

Page range: 19 - 44

Received: Jan 11, 2023

Accepted: Apr 18, 2023

DOI: https://doi.org/10.2478/auom-2024-0002

Keywordspolynomial ring, additive cyclic codes, Gray map

© 2024 Mohammad Ashraf et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

The study of ℤ_pℤ_p[u, v]-additive cyclic codes and their application in obtaining Optimal and MDSS codes

Keywords
polynomial ring, additive cyclic codes, Gray map