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By constructing a Gray map, constacyclic codes of arbitrary lengths over ring \(R = Z_{p^m } + vZ_{p^m }\) are studied, where v 2 = v The structure of constacyclic codes over R and their dual codes are obtained. A necessary and sufficient condition for a linear code to be self-dual constacyclic is given. In particular, (1+( v +1) αp )-constacyclic codes over R are classified in terms of generator polynomial, where α is a unit of \(Z_{p^m }\) .
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