Constacyclic codes as invariant subspaces
A constacyclic code (sometimes also referred to as pseudo-cyclic code) is a linear code over a field GF\((q)\) invariant under the permutation \(\psi_a:\) \((c_0,c_1\ldots,c_{n-1})\mapsto (ac_{n-1},c_0,c_1\ldots,c_{n-1})\) (for a fixed field element \(a\)). (In the introduction of the article the definition is not correct). The matrix \(A(n,a)\) of \(\psi_a\) (with respect to the standard basis) fullfils the equation \(A(n,a)^{-1}=A(n,a^{-1})^t\) (here, there is a misprint). In this article, generalizations of results known for cyclic codes are derived for constacyclic codes \(\mathcal C\), partly by writing \(\mathcal C\) as direct sum of some mini\-mal \(\psi_a-\)invariant subspaces \(U_i\) and using linear algebra. For instance there are defined primitive idempotent matrices fixing an \(U_i\), anihilating the other \(U_j\), and having as sum a matrix fixing exactly \(\mathcal C\). Furthermore, the authors prove a generalization of the BCH-bound and of the Hartmann-Tzeng-Roos bound [see \textit{C. Roos}, J. Comb. Theory, Ser. A 33, 229--232 (1982; Zbl 0497.94011)] for the minimal distance of constacyclic codes and apply this bound to an example of a constacyclic code of parameters \(n=25\), \(q=7\), \(a=-1\) showing that it has minimum distance \(\geq 6\).
scientific article; zbMATH DE number 5382426