Cyclic codes over finite rings (Q1377884)

A linear left code \(C\) of length \(n\) over a finite ring \(R\) is a submodule of \({}_R R^n\). It is called splitting if it is a direct summand of \({}_R R^n\). \(C\) is a cyclic linear left code if it is a left ideal of \(R[x]/(x^n -1)\) and it is called splitting if it is a direct summand of \({}_R (R[x]/(x^n -1))\). The main result is a complete characterization of cyclic splitting codes as follows: Theorem: For a cyclic linear left code of length \(n\) over a finite ring \(R\) the following are equivalent: (a) \(C\) is a splitting code. (b) There exists a divisor \(g\) of \(x^n -1\) in \(R[x]\) such that \(C=R[x]g/(x^n -1)\).