Some classes of quasi-twisted codes over finite chain rings

Abstract

Let R be an arbitrary commutative finite chain ring and \(\gamma \) a fixed generator of the maximal ideal of R. Suppose s is the nilpotency index of \(\gamma \) and F is the residue field of R modulo its ideal \(\gamma R\), i.e. \(F = R / \gamma R\), \(\vert F \vert = q\) with \(q=p^{\alpha }\) for some prime number p and let \(R^{\times }\) denote the multiplicative group of units of R. For any \(\omega \in R^{\times }\) and \(t \ge \lceil \frac{s}{2}\rceil \), the structural properties and dual codes of \((1+ \omega \gamma ^{t})\)-quasi-twisted (QT) codes of length \(n=\ell m\), with \((m, p)=1\), over R are given. The key idea is to view a \((1 + \omega \gamma ^{t})\)-QT code over R as a linear code over \(R_{m} = R[x] / \langle x^{m} - (1 + \omega \gamma ^{t}) \rangle \). Furthermore, given the decomposition of a \((1+ \omega \gamma ^{t})\)-QT code, we provide the decomposition of its dual code. As a result, a characterization of self-dual \((1+ \omega \gamma ^{t})\)-QT codes over a finite chain ring R, with \((1+ \omega \gamma ^{t})=(1+ \omega \gamma ^{t})^{-1}\), is provided. By using the Chinese remainder theorem or the discrete Fourier transform, the ring \(R[x] / \langle x^{m} - (1 + \omega \gamma ^{t}) \rangle \) can be decomposed into a direct sum of finite chain rings. The inverse transform of the discrete Fourier transform produces a method for deriving \((1+ \omega \gamma ^{t})\)-QT codes from codes of lower lengths.