Abstract
In this paper, we study \(\lambda \)-constacyclic codes over the ring \(R=\mathbb {Z}_4+u\mathbb {Z}_4\) where \(u^{2}=1\), for \(\lambda =3+2u\) and \(2+3u\). Two new Gray maps from R to \(\mathbb {Z}_4^{3}\) are defined with the goal of obtaining new linear codes over \(\mathbb {Z}_4\). The Gray images of \(\lambda \)-constacyclic codes over R are determined. We then conducted a computer search and obtained many \(\lambda \)-constacyclic codes over R whose \(\mathbb {Z}_4\)-images have better parameters than currently best-known linear codes over \(\mathbb {Z}_4\).
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References
Ashraf M., Mohammed G.: \((1+2u)\)-constacyclic codes over \(\mathbb{Z}_4+u\mathbb{Z}_4\). arXiv:1504.03445v1.
Aydin N., Asamov T.: A database of \(\mathbb{Z}_4\) codes. J. Comb. Inf. Syst. Sci. 34, 1–12 (2009).
Aydin N., Dertli A., Cengellenmis Y.: Cyclic and constacyclic codes over \(\mathbb{Z}_4+w\mathbb{Z}_4\). Pre-print.
Bandi R., Bhaintwal M.: Self dual codes over \(\mathbb{Z}_4+w\mathbb{Z}_4\). Discret. Math. Alg. Appl. 7, 1550014 (2015).
Bandi R., Bhaintwal M.: Negacyclic codes over \(\mathbb{Z}_4+u\mathbb{Z}_4\). arXiv:1412.3751v1.
Bandi R., Bhaintwal M.: Cyclic codes over \(\mathbb{Z}_4+u\mathbb{Z}_4\). arXiv:1501.01327.
Bonnecaze A., Rains A.M., Solé P.: 3-Colored 5-designs and \(\mathbb{Z}_4\)-codes. J. Stat. Plan. Inf. 86, 349–368 (2000).
Database of \(\mathbb{Z}_4\) codes [online]. Z4Codes.info (Accessed February 2017).
Gao J., Gao Y., Fu F.: On linear codes over \(\mathbb{Z}_4+v\mathbb{Z}_4\). arXiv:1402.6771v2.
Hammons Jr. A.R., Kumar P.V., Calderbank A.R., Solane N.J.A., Solé P.: The \(\mathbb{Z}_4\) linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inf. Theory 40, 301–319 (1994).
Martinez-Moro E., Szabo S.: On codes over local Frobenius non-chain rings of order 16. Contemp. Math. 634, 227–241 (2015).
Martinez-Moro E., Szabo S., Yildiz B.: Linear codes over \(\mathbb{Z}_4[x]/\langle x^{2}+2x\rangle \). Int. J. Inf. Coding Theory 2, 78–96 (2015).
Ozen M., Ozzaim T., Aydin N.: Cyclic codes over \(\mathbb{Z}_4 + u\mathbb{Z}_4 + u^2\mathbb{Z}_4\). Turk. J. Math. (to appear)
Ozen M., Uzekmek F.Z., Aydin N., Özzaim N.T.: Cyclic and some constacyclic codes over the ring \(\mathbb{Z}_4[u]/\langle u^2-1 \rangle \). Finite Fields Appl. 38, 27–39 (2016).
Pattanayak S., Singh A.: A class of cyclic codes over the ring \(Z_{4}[u]/\langle u^{2}\rangle \) and its Gray image. arXiv:1507.04938v1.
Shi M., Qian L., Sok L., Aydin N., Solé P.: On constacyclic codes over \(\mathbb{Z}_4[u]/\langle u^2-1\rangle \). Finite Fields Appl. 45, 86–95 (2017).
Shi M., Qian L., Sok L.: On constacyclic codes over \(\mathbb{Z}_4\left[u\right] /\langle u^{2}-1\rangle \) and their Gray images. arXiv:1608.00820.
Solé P.: A quaternary cyclic code, and a family of quadriphase sequences with low correlation properties. In: Springer Lecture Notes in Computer Science, vol. 388, pp. 193–201 (1998).
Wan Z.X.: Quaternary Codes. World Scientific, Singapore (1997).
Yildiz B., Aydin N.: On cyclic codes over \(\mathbb{Z}_4+u\mathbb{Z}_4\) and their \(\mathbb{Z}_4\) images. Int. J. Inf. Coding Theory 2, 226–237 (2014).
Yildiz B., Karadeniz S.: Linear codes over \(\mathbb{Z}_4+u\mathbb{Z}_4\), Macwilliams identities, projections and formally self dual codes. Finite Fields Appl. 27, 24–40 (2014).
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Communicated by T. Helleseth.
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Aydin, N., Cengellenmis, Y. & Dertli, A. On some constacyclic codes over \(\mathbb {Z}_{4}\left[ u\right] /\left\langle u^{2}-1\right\rangle \), their \(\mathbb {Z}_4\) images, and new codes. Des. Codes Cryptogr. 86, 1249–1255 (2018). https://doi.org/10.1007/s10623-017-0392-y
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DOI: https://doi.org/10.1007/s10623-017-0392-y