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Erschienen in:
01.02.2016 | Original Research
Some results on quasi-twisted codes over \(\mathbb {F}_2[u]/\left( u^{k+1}\right) \)
Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2016
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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.