On additive conjucyclic codes over fields and rings of order <inline-formula><tex-math id="M1">$ p^{2} $</tex-math></inline-formula> using Gaussian integers

In this paper, structures of conjucyclic codes and their duals over fields and rings of order $ p^{2} $ were studied. The approach was based on realizing additive conjucyclic codes over quotient rings of Gaussian integers, i.e., $ R_{p} = \mathbb{Z}\left[ i\right] /\langle p\rangle $, which has been proven to be an effective choice compared to previous studies on conjucyclic codes. We show that the algebraic structures of these codes are related to those algebraic structures of linear cyclic and linear negacyclic codes over finite fields $ \mathbb{F}_{p} $ of order $ p. $ We also introduce a definition of inner product on $ R_{p} $ and show that duals of additive conjucyclic codes over $ R_{p} $ based on this inner product are also additive conjucyclic codes. As a result, we show that there are no additive self-dual conjucyclic codes over $ R_{p}. $ We then study the properties and structures of additive complementary dual (ACD) conjucyclic codes and maximum distance separable (MDS) conjucyclic codes over $ R_{p}. $ As an application of our study, we construct classes of conjucyclic codes that consist of both ACD and MDS conjucyclic codes with parameters $ \left( n, p^{4k}, n-2k+1\right) . $ Furthermore, we provide many examples of optimal additive conjucyclic codes with the largest Hamming distance among additive conjucyclic codes over $ R_{p}. $