Abstract
The homogeneous weight (metric) is useful in the construction of codes over a ring of integers \(\mathbb {Z}_{p^l}\) (p prime and \(l \ge 1\) an integer). It becomes Hamming weight when the ring is taken to be a finite field and becomes Lee weight when the ring is taken to be \(\mathbb {Z}_{4}\). This paper presents homogeneous weight distribution and total homogeneous weight of burst and repeated burst errors in the code space of n-tuples over \(\mathbb {Z}_{p^l}\). Necessary and sufficient conditions for existence of an (n, k) linear code over \(\mathbb {Z}_{p^l}\) correcting the error patterns with respect to the homogeneous weight are derived.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Calderbank, A.R., Sloane, N.J.A.: Modular and p-adic cyclic codes. Des. Codes Cryptogr. 6, 21–35 (1995)
Constantinescu, I., Heise, W.: A metric for codes over residue class rings of integers. Problemy Peredachi Informatsii 33(3), 22–28 (1997)
Berardi, L., Dass, B.K., Verma, R.: On 2-repeated burst error detecting codes. J. Stat. Theory Pract. 3(2), 381–391 (2009)
Dass, B.K., Verma, R.: Repeated burst error correcting linear codes. Asian-Eur. J. Math. 1(4), 303–335 (2008)
Fire, P.: A class of multiple-error-correcting binary codes for non-independent errors, Sylvania Report, RSL-E-2, Sylvania Reconnaissance Systems Laboratory. Calif, Mountain View (1959)
Greferath, M., Schmidt, S.E.: Gray isometries for finite chain rings and a nonlinear ternary (36,312,15) code. IEEE Trans. Inf. Theory 45, 2522–2524 (1999)
Greferath, M., Schmidt, S.E.: Finite-ring combinatorics and MacWilliams equivalence theorem. J. Comb. Theory(A) 92(1), 17–28 (2000)
Greferath, M., Nechaev, A., Wisbauer, R.: Finite quasi-Frobenius modules and linear codes. J. Algebra its Appl. 3(3), 247–272 (2004)
Hamming, R.W.: Error-detecting and error-correcting codes. Bell Syst. Tech. J. 29(2), 147–160 (1950)
Lee, C.Y.: Some properties of non-binary error correcting codes. IEEE Trans. Inf. Theory IT–4, 77–82 (1958)
Li, W.S., Wan, L.Y., Yue, M.T., Chen, W., Zhang, X.D.: Linear codes over the finite ring \(Z_{15}\). Adv. Linear Algebra Matrix Theory 10, 1–5 (2020)
Moeini, M., Rezaei, R., Samei, K.: MacWilliams identity for linear codes over finite chain rings with respect to homogeneous weight. J. Korean Math. Soc. 58(5), 1163–1173 (2021)
Peterson, W.W., Weldon, E.J.: Error correcting codes, 2nd edn. MIT Press, Cambridge, Massachusetts (1972)
Sharma, B.D., Dass, B.K.: On weight of burst, Indian Journal of. Pure Appl. Math. 8(12), 1519–1524 (1977)
Sharma, B.D., Rohtagi, B.: Some results on weights of vectors having \(m\)-repeated bursts. Cybern. Inf. Technol. 11(3), 3–11 (2011)
Srinivas, K.V., Jain, R., Saurav, S., Sikdar, S.K.: Small-world network topology of hippocampal neuronal network is lost, in an in vitro glutamate injury modelof epilepsy. Eur. J. Neurosci. 25, 3276–3286 (2007)
Temiz, F., Siap, V.: Linear block and array codes correcting repeated CT-burst errors. Albanian J. Math. 7(2), 77–92 (2013)
Yildiz, B., Abdljabbar, M.A.: On \(p\)-ary local Frobenius rings and their homogeneous weights. Appl. Math. Inf. Sci. 10(6), 2109–2116 (2016)
Yildiz, B., Ozger, Z.O.: A generalization of the Lee weight \(\mathbb{Z} _{p^k}\). TWMS J. Appl. Eng. Math. 2(2), 145–153 (2012)
Yildiz, B., Tufekci, N.: Optimal divisible binary codes from Gray-Homogeneous images of codes over \(R_{k, m}\). Eur. J. Pure Appl. Math. 10(5), 1112–1123 (2017)
Acknowledgements
This paper is a part of Ph.D. thesis submitted by the second author to the University of Delhi, 2021.
Funding
None.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Das, P.K., Kumar, S. Linear Codes Correcting Repeated Bursts Equipped with Homogeneous Distance. Theory Comput Syst (2024). https://doi.org/10.1007/s00224-024-10166-y
Accepted:
Published:
DOI: https://doi.org/10.1007/s00224-024-10166-y