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arXiv · 1608.00820

On constacyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$ and their Gray images

Minjia Shi·Liqing Qian·Lin Sok·Nuh Aydin·Patrick Solé
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Abstract

We first define a new Gray map from $R=\mathbb{Z}_4+u\mathbb{Z}_4$ to $\mathbb{Z}^{2}_{4}$, where $u^2=1$ and study $(1+2u)$-constacyclic codes over $R$. Also of interest are some properties of $(1+2u)$-constacyclic codes over $R$. Considering their $\mathbb{Z}_4$ images, we prove that the Gray images of $(1+2u)$-constacyclic codes of length $n$ over $R$ are cyclic codes of length $2n$ over $\mathbb{Z}_4$. In many cases the latter codes have better parameters than those in the online database of Aydin and Asamov. We also give a corrected version of a table of new cyclic $R$-codes published by Özen et al. in Finite Fields and Their Applications, {\bf 38}, (2016) 27-39.

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Minjia Shi, Liqing Qian, Lin Sok, Nuh Aydin, Patrick Solé. 2016-12-25. On constacyclic codes over $\mathbb{Z}_4[u]/\langle u^2-1\rangle$ and their Gray images. https://doi.org/10.1016/j.ffa.2016.11.016

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