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Designs, Codes and Cryptography

, Volume 86, Issue 4, pp 841–859 | Cite as

New constructions of MDS symbol-pair codes

  • Baokun Ding
  • Gennian GeEmail author
  • Jun Zhang
  • Tao Zhang
  • Yiwei Zhang
Article

Abstract

Motivated by the application of high-density data storage technologies, symbol-pair codes are proposed to protect against pair-errors in symbol-pair channels, whose outputs are overlapping pairs of symbols. The research of symbol-pair codes with the largest minimum pair-distance is interesting since such codes have the best possible error-correcting capability. A symbol-pair code attaining the maximal minimum pair-distance is called a maximum distance separable (MDS) symbol-pair code. In this paper, we focus on constructing linear MDS symbol-pair codes over the finite field \({\mathbb {F}}_{q}\). We show that a linear MDS symbol-pair code over \({\mathbb {F}}_{q}\) with pair-distance 5 exists if and only if the length n ranges from 5 to \(q^2+q+1\). As for codes with pair-distance 6, length ranging from \(q+2\) to \(q^{2}\), we construct linear MDS symbol-pair codes by using a configuration called ovoid in projective geometry. With the help of elliptic curves, we present a construction of linear MDS symbol-pair codes for any pair-distance \(d+2\) with length n satisfying \(7\le d+2\le n\le q+\lfloor 2\sqrt{q}\rfloor +\delta (q)-3\), where \(\delta (q)=0\) or 1.

Keywords

Symbol-pair read channels MDS symbol-pair codes Projective geometry Elliptic curves 

Mathematics Subject Classification

94B25 94B60 

Notes

Acknowledgements

The authors express their gratitude to the two anonymous reviewers for their detailed and constructive comments which are very helpful to the improvement of this paper, and to Prof. Tuvi Etzion, the Associate Editor, for his insightful advice and excellent editorial job. The research of Gennian Ge is supported by the National Natural Science Foundation of China under Grant Nos. 11431003 and 61571310, Beijing Hundreds of Leading Talents Training Project of Science and Technology, and Beijing Municipal Natural Science Foundation. The research of Jun Zhang is supported by the National Natural Science Foundation of China under Grant No. 11601350, by Scientific Research Project of Beijing Municipal Education Commission under Grant No. KM201710028001, and by Beijing outstanding talent training program under Grant No. 2014000020124G140. Jun Zhang is supported by Chinese Scholarship Council during visiting the University of Oklahoma, USA.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Baokun Ding
    • 1
  • Gennian Ge
    • 2
    Email author
  • Jun Zhang
    • 3
  • Tao Zhang
    • 1
  • Yiwei Zhang
    • 3
  1. 1.School of Mathematical SciencesZhejiang UniversityHangzhouChina
  2. 2.School of Mathematics and Information ScienceGuangzhou UniversityGuangzhouChina
  3. 3.School of Mathematical SciencesCapital Normal UniversityBeijingChina

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