Primitive idempotent tables of cyclic and constacyclic codes

Abstract

For any (Formula presented.) a (Formula presented.)-constacyclic code (Formula presented.), of length (Formula presented.) is a set of polynomials in the ring (Formula presented.), which is generated by some polynomial divisor (Formula presented.) of (Formula presented.). In this paper a general expression is presented for the uniquely determined idempotent generator of such a code. In particular, if (Formula presented.), where (Formula presented.) is an irreducible factor polynomial of (Formula presented.), 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 (Formula presented.) with (Formula presented.) 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 (Formula presented.) (cyclic codes) and (Formula presented.) (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).