Construction of linear codes over $\mathfrak{R}=\sum_{s=0}^{4} v_{5}^{s}\mathcal{A}_{4}$

The aim of this paper is to propose a new family of codes.  We define this family over the ring $\mathfrak{R}=\sum_{s=0}^{4} v_{5}^{s}\mathcal{A}_{4}$, with $v_{5}^{5}=v_{5}$.  We derive its properties, a generator matrix and Gray images.  This new family of codes is illustrated by three applications.

  1. Chatouh K., Guenda K., Aaron Gulliver T.  New classes of codes over $R_{q,p,m}=\mathbb{Z}_{p^{m}}[u_{1},u_{2},\cdots,u_{q}]/\langle u_{i}^{2}=0,u_{i}u_{j}-u_{j}u_{i}\rangle$ and their applications.  Computational and Applied Mathematics.  39, 152 (2020).
  2. Yildiz B., Karadeniz S.  Linear codes over  $\mathbb{Z}_{4}+u\mathbb{Z}_{4}$: MacWilliams identities, projections, formally self-dual codes.  Finite Fields and Their Applications.  27, 24–40 (2014).
  3. Klemm M.  Selbstduale Codes über dem Ring der ganzen Zahlen modulo 4.  Archiv der Mathematik.  53 (2), 201–207 (1989).
  4. Hammons A. R., Kumar P. V., Calderbank A. R., Sloane N. J. A., Solé P.  The $\mathbb{Z}_{4}$-linearity of Kerdock, Preparata, Goethals and related codes.  IEEE Transactions on Information Theory.  40 (2), 301–319 (1994).
  5. Hammons A. R., Kumar P. V., Calderbank A. R., Sloane N. J. A., Solé P.  On the apparent duality of the Kerdock and Preparata codes.  International Symposium on Applied Algebra, Algebraic Algorithms and Error–Correcting Codes. 13–24 (1993).
  6. Bustomi, Santika A. P., Suprijanto D.  Linear codes over the ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+v\mathbb{Z}_{4}+w\mathbb{Z}_{4}+uv\mathbb{Z}_{4}+uw\mathbb{Z}_{4}+vw\mathbb{Z}_{4}+uvw\mathbb{Z}_{4}$.  Preprint arXiv:1904.11117v1 (2019).
  7. Li P., Guo X., Zhu S., Kai X.  Some results on linear codes over the ring $\mathbb{Z}_{4}+u\mathbb{Z}_{4}+v\mathbb{Z}_{4}+uv\mathbb{Z}_{4}$.  Journal of Applied Mathematics and Computing.  54 (1–2), 307-324 (2017).
  8. Ndiaye O.  One cyclic codes over $F_{p^{k}}+vF_{p^{k}}+v^{2}F_{p^{k}}+\ldots+v^{r}F_{p^{k}}$.  Gulf Journal of Mathematics.  4 (4), (2016).
  9. Liu Y., Shi M., Solé P.  Quadratic Residue Codes over $\mathbb{F}_{p}+v\mathbb{F}_{p}+v^{2}\mathbb{F}_{p}$.  International Workshop on the Arithmetic of Finite Fields. 204–211 (2014).
  10. Qian J.-F., Zhang L.-N., Zhu S.-X.  Cyclic Codes over $\mathbb{F}_{p}+ u\mathbb{F}_{p}+\ldots+ u^{k-1}\mathbb{F}_{p}$.  IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences.  E88-A (3), 795–797 (2005).
  11. Aydin N., Ray-Chaudhuri D. K.  Quasi-cyclic codes over $\mathbb{Z}_{4}$ and some new binary codes.  IEEE Transactions on Information Theory.  48 (7), 2065–2069 (2002).
  12. Gao J., Shi M., Wu T., Fu F.-W.  On double cyclic codes over $\mathbb{Z}_{4}$.  Finite Fields and Their Applications.  39, 233–250 (2016).
  13. Melakhessou A., Guenda K., Gulliver T. A., Shi M., Solé P.  On Codes over $\mathbb{F}_{q}+v\mathbb{F}_{q}+v^{2}\mathbb{F}_{q}$.  Journal of Applied Mathematics and Computing.  57 (1), 375–391 (2018).
  14. Grassl M.  Bounds on the minimum distance of linear codes and quantum codes.  Online available at www.codetables.de. Accessed on 2021/08/20 (2019).
Mathematical Modeling and Computing, Vol. 10, No. 1, pp. 147–158 (2023)