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

Let S = ℤ_p[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 ℤ_pS-additive cyclic and constacyclic codes. The minimal polynomials and spanning sets of ℤ_pS-additive cyclic and constacyclic codes are also determined. These codes are identified as S[y]-submodules of the ring S_{β1, β2} = ℤ_p[y]/〈y^β₁ − 1〉 × S[y]/〈y^β₂ − 1〉. Some results that represent the relationship between the minimal polynomials of ℤ_pS-additive cyclic codes and their duals have been obtained. Furthermore, optimal ℤ_pS-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.