Abstract
Recently, Jia proposed the decompositions and trace representations of quasi-twisted (QT) codes over finite fields (Finite Fields Appl 18:237–257, 2012). The present paper can be viewed as a complementary part of Jia’s work. We investigate some other useful properties of \(\lambda \)-QT codes over finite fields, including the lower Hamming distance bounds, enumerations and searching algorithm for generators. As an interesting application of \(\lambda \)-QT codes over finite fields, we study \(\lambda \)-QT codes over the finite non-chain ring \(\mathbb {F}_q+v\mathbb {F}_q\) briefly.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aydin, N., Siap, I., Ray-Chaudhuri, D.: The structure of \(1\)-generator quasi-twisted codes and new linear codes. Des. Codes Cryptogr. 24, 313–326 (2001)
Bhaintwal, M., Wasan, S.: On quasi-cyclic codes over \(\mathbb{Z}_q\). Appl. Algebra Eng. Commun. Comput. 20, 459–480 (2009)
Cao, Y.: \(1\)-generator quasi-cyclic codes over finite chain rings. Appl. Algebra Eng. Commun. Comput. 24, 53–72 (2013)
Daskalov, R., Abualrub, T.: New quasi-twisted quaternary linear codes. IEEE Trans. Inform. Theory 46, 2642–2643 (2000)
Dinh, H.: Cyclic and negacyclic codes over finite chain rings. IEEE Trans. Inform. Theory 50, 1728–1743 (2004)
Gao, J., Kong, Q.: 1-generator quasi-cyclic codes over \(\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+\ldots +u^{s-1}\mathbb{F}_{p^m}\). J. Frank. Inst. 350, 3260–3276 (2013)
Gao, J., Kong, Q.: One generator \((1+u)\)-quasi twisted codes over \(\mathbb{F}_2+u\mathbb{F}_2\). Math. Comput. 2, 1–5 (2013)
Güneri, C., Özbudak, F.: A bound on the minimum distance of quasi-cyclic codes. SIAM J. Discret. Math. 26, 1781–1796 (2012)
Hammons, A., Kumar, P., Calderbank, A., Sloane, N., Solé, P.: The \(\mathbb{Z}_4\)-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inform. Theory 40, 301–319 (1994)
Jia, Y.: On quasi-twisted codes over finite fields. Finite Fields Appl. 18, 237–257 (2012)
Ling, S., Solé, P.: On the algebra structure of quasi-cyclic codes I: finite fields. IEEE Trans. Inform. Theory 47, 2751–2760 (2003)
Wan, Z.-X.: Algebra and Coding, 3rd edn. Higher Education Press, Beijing (1997). (in Chinese)
Zhu, S., Wang, Y., Shi, M.: Some results on cyclic codes over \(\mathbb{F}_2+v\mathbb{F}_2\). IEEE Trans. Inform. Theory 56, 1680–1684 (2010)
Zhu, S., Wang, L.: A class of constacyclic codes over \(\mathbb{F}_p+v\mathbb{F}_p\). Discret. Math. 311, 2677–2682 (2011)
Acknowledgments
The authors are deeply indebted to the referees and wish to thank them for their important suggestions and comments. This research is supported by the National Key Basic Research Program of China (Grant No. 2013CB834204), and the National Natural Science Foundation of China (Grant Nos. 61171082, 61301137).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gao, J., Fu, FW. Note on quasi-twisted codes and an application. J. Appl. Math. Comput. 47, 487–506 (2015). https://doi.org/10.1007/s12190-014-0787-0
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-014-0787-0