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For any \( \lambda \in GF(q)^{\ast} \) a \( \lambda \)-constacyclic code \( C^{n,q,\lambda } : = \,\langle g(x) \rangle \), of length \( n \) is a set of polynomials in the ring \( GF(q)[x]/x^{n} - \lambda \), which is generated by some polynomial divisor \( g(x) \) of \( x^{n} - \lambda \). In this paper a general expression is presented for the uniquely determined idempotent generator of such a code. In particular, if \( g(x): = (x^{n} - \lambda) / P_{t}^{n,q,\lambda } (x) \), where \( P_{t}^{n,q,\lambda } (x) \) is an irreducible factor polynomial of \( x^{n} - \lambda \), one obtains a so-called minimal or irreducible constacyclic code. The idempotent generator of a minimal code is called a primitive idempotent generating polynomial or, shortly, a primitive idempotent. It is proven that for any triple \( (n,q,\lambda ) \) with \( (n,q) = 1 \) the set of primitive idempotents gives rise to an orthogonal matrix. This matrix is closely related to a table which shows some resemblance with irreducible character tables of finite groups. The cases \( \lambda = 1 \) (cyclic codes) and \( \lambda = - 1 \) (negacyclic codes), which show this resemblance most clearly, are studied in more detail. All results in this paper are extensions and generalizations of those in van Zanten (Des Codes Cryptogr 75:315–334, 2015).

Primitive idempotent tables of cyclic and constacyclic codes