Non-invertible-element constacyclic codes over finite PIRs

In this paper we introduce the notion of λ-constacyclic codes over finite rings R for arbitrary element λ R. We study non-invertible-element constacyclic (NIE-constacyclic codes) principal ideal (PIRs). determine algebraic structures all NIE-constacyclic chain rings, give unique form sets defining polynomials and obtain their minimum Hamming distances. A general duals is also provided. particular, a necessary sufficient condition dual an code to be code. Using Chinese Remainder Theorem, PIRs. Furthermore, construct some optimal PIRs in sense that they achieve maximum possible distances given lengths cardinalities.