Abstract
We construct two series of linear codes C(G) over \(\mathbb {F}_{q}[x]/(x^2)\) and \(GR(p^2,m)\) reaching the Griesmer bound. Moreover, we consider the Gray images of C(G). The results show that the Gray images of C(G) over \(\mathbb {F}_{q}[x]/(x^2)\) are linear and also reach the Griesmer bound in some cases, and many of linear codes over \(\mathbb {F}_{q}\) we constructed have two Hamming (non-zero) weights.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Greferath M., Schmidt S.E.: Gray isometries for finite chain rings and a nonlinear ternary \((36,3^{12},5)\) code. IEEE Trans. Inf. Theory 45(7), 2522–2524 (1999).
Griesmer J.H.: A bound for error correcting codes. IBM J. Res. Dev. 4(5), 532–542 (1960).
Hammons Jr. 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 Trans. Inf. Theory 40, 301–319 (1994).
Klein A.: On codes meeting the Griesmer bound. Discret. Math. 274(1–3), 289–297 (2004).
Shiromoto K., Storme L.: A Griesmer bound for linear codes over finite quasi-Frobenius rings. Discret. Appl. Math. 128(1), 263–274 (2003).
Solomon G., Stiffler J.J.: Algebraically punctured cyclic codes. Inf. Control 8(2), 170–179 (1965).
Tamari F.: A construction of some \([n, k, d]_q\) codes meeting the Griesmer bound. Discret. Math. 116(1–3), 269–287 (1993).
Voloch J.F., Walker J.L.: Homogeneous weights and exponential sums. Finite Fields Their Appl. 9(3), 310–321 (2003).
Wan Z.: Lecture Notes on Finite Fields and Galois Rings. World Scientific, Singapore (2003).
Yildiz B.: A combinatorial construction of the Gray map over Galois rings. Discret. Math. 309(10), 3408–3412 (2009).
Acknowledgements
The work of J. Li was supported by National Natural Science Foundation of China(NSFC) under Grant 11501156 and the Anhui Provincial Natural Science Foundation under Grant 1508085SQA198. The work of A. Zhang was supported by NSFC under Grant 11401468. The work of K. Feng was supported by NSFC under Grant Nos. 11471178, 11571007 and the Tsinghua National Lab. for Information Science and Technology.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J.-L. Kim.
Rights and permissions
About this article
Cite this article
Li, J., Zhang, A. & Feng, K. Linear codes over \(\mathbb {F}_{q}[x]/(x^2)\) and \(GR(p^2,m)\) reaching the Griesmer bound. Des. Codes Cryptogr. 86, 2837–2855 (2018). https://doi.org/10.1007/s10623-018-0479-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-018-0479-0