Constacyclic and quasi-twisted codes over \(\mathbb{Z}_q [u] / \langle u^2 - 1 \rangle\) and new \(\mathbb{Z}_4\)-linear codes (Q6605896)

This paper studies different classes of codes over the ring \(R = \mathbb{Z}_q + u\mathbb{Z}_q,\) with \(u^2=1\) and \(q=p^m\) for a prime number \(p.\) A linear code \(C\) of length \(n\) over \(R\) is an \(R-\)submodule of \(R^n. \) The constacyclic shift on \(R^n\) is defined as \(\rho_\lambda(c_0, \ldots, c_{n-1}) = (\lambda c_{n-1}, c_0, \ldots, c_{n-2}),\) where \(\lambda\) is a unit in \(R.\) A linear code \(C\) is called constacyclic if \(\rho_\lambda (C) = C.\) Quasi-twisted (QT) codes are a generalization of constacyclic codes that contain the class of quasi-cyclic (QC) codes as a special case. QT code of index \(\ell\) over R is a code \(C\) such that \(\tau_{\ell,\lambda}(C) = C,\) where \(\tau_{\ell,\lambda}(c_0,\ldots,c_{n-1})=(\lambda c_{n-\ell}, \ldots, \lambda c_{n-1}, c_{0}, \ldots, c_{n-\ell-1}).\)\N\NSome structural properties are determined, and a decomposition of QT codes over \(R\) of length \(n\) and index \(\ell\) is given. The authors prove that the image of a constacylic code over R is a QT code over \(\mathbb{Z}_q.\) Also, it's shown that the dual of a QT code is a QT code of the same length and index.\N\NBy decomposing \(R\) into a product of local rings, it's proven that the polynomial \(x^s-\lambda\) factors into pairwise coprime monic irreducible polynomials over \(R.\) The linear complementary pair (LCP) codes \((C, D),\) where \(C, D\) are both constacyclic codes, are taken into consideration as an application of these results. Additionally, the authors have found 116 new linear codes over \(\mathbb{Z}_4\) from the Gray images of QT codes over the same ring for \(q=4.\)