On some constacyclic codes over $$\mathbb {Z}_{4}\left[ u\right] /\left\langle u^{2}-1\right\rangle $$Z4u/u2-1, their $$\mathbb {Z}_4$$Z4 images, and new codes

In this paper, we study $$\lambda $$λ-constacyclic codes over the ring $$R=\mathbb {Z}_4+u\mathbb {Z}_4$$R=Z4+uZ4 where $$u^{2}=1$$u2=1, for $$\lambda =3+2u$$λ=3+2u and $$2+3u$$2+3u. Two new Gray maps from R to $$\mathbb {Z}_4^{3}$$Z43 are defined with the goal of obtaining new linear codes over $$\mathbb {Z}_4$$Z4. The Gray images of $$\lambda $$λ-constacyclic codes over R are determined. We then conducted a computer search and obtained many $$\lambda $$λ-constacyclic codes over R whose $$\mathbb {Z}_4$$Z4-images have better parameters than currently best-known linear codes over $$\mathbb {Z}_4$$Z4.