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Designs, Codes and Cryptography

, Volume 87, Issue 4, pp 855–863 | Cite as

Codes correcting restricted errors

  • Igor E. Shparlinski
  • Arne WinterhofEmail author
Article
  • 49 Downloads
Part of the following topical collections:
  1. Special Issue: Finite Geometries

Abstract

We study the largest possible length B of \((B-1)\)-dimensional linear codes over \(\mathbb {F}_q\) which can correct up to t errors taken from a restricted set \({\mathcal {A}}\subseteq \mathbb {F}_q^*\). Such codes can be applied to multilevel flash memories. Moreover, in the case that \(q=p\) is a prime and the errors are limited by a constant we show that often the primitive \(\ell \)th roots of unity, where \(\ell \) is a prime divisor of \(p-1\), define good such codes.

Keywords

Linear codes Restricted errors Packing sets 

Mathematics Subject Classification

68P30 94B05 94B65 

Notes

Acknowledgements

This work was initiated during a visit of the authors to the Paris Lodron University of Salzburg. The authors are grateful for its support. During the preparation of this work Igor E. Shparlinski was supported by the ARC Grants DP170100786 and DP180100201, and Arne Winterhof was supported by the Austrian Science Fund FWF Project P 30405-N32. The authors like to thank the anonymous referees for their valuable suggestions.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
  2. 2.Johann Radon Institute for Computational and Applied MathematicsAustrian Academy of SciencesLinzAustria

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