Some results on quasi-twisted codes over $\mathbb {F}_2[u]/\left( u^{k+1}\right)$

Abstract

$\left( 1+u^k \right)$-quasi-twisted codes over the ring $R=\mathbb {F}_{2}[u]/\left( u^{k+1}\right)$ are introduced. The key idea is to consider a $(1+u^k)$-quasi-twisted code over $R$ as a linear code over $R_{m}=R[x]/\left( x^m+\left( 1+u^k \right) \right)$. The dual of $(1+u^k)$-quasi-twisted codes are also studied. By using the Chinese remainder theorem or the discrete Fourier transform, the ring $R[x]/\left( x^m+\left( 1+u^k \right) \right)$ can be decomposed into a direct sum of finite chain rings. The inverse transform of the discrete Fourier transform induces a $\left( 1+u^k \right)$-quasi-twisted code construction from codes of lower lengths.