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International Symposium on Algorithms and Computation

ISAAC 2013: Algorithms and Computation pp 100–110Cite as

On Determining Deep Holes of Generalized Reed-Solomon Codes

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8283))

Abstract

For a linear code, deep holes are defined to be vectors that are further away from codewords than all other vectors. The problem of deciding whether a received word is a deep hole for generalized Reed-Solomon codes is proved to be co-NP-complete [9][5]. For the extended Reed-Solomon codes \(RS_q(\mathbb{F}_q,k)\), a conjecture was made to classify deep holes in [5]. Since then a lot of effort has been made to prove the conjecture, or its various forms. In this paper, we classify deep holes completely for generalized Reed-Solomon codes RS p (D,k), where p is a prime, \(|D| > k \geqslant \frac{p-1}{2}\). Our techniques are built on the idea of deep hole trees, and several results concerning the Erdös-Heilbronn conjecture.

The research is partially supported by 973 Program (Grant 2013CB834201) and NSF under grants CCF-0830522 and CCF-0830524 for Q.C. and J.Z., and by National Science Foundation of China (11001170) and Ky and Yu-Fen Fan Fund Travel Grant from the AMS for J.L.. The full version appears as arXiv:1309.3546.

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Author information

Authors and Affiliations

  1. School of Computer Science, The University of Oklahoma, Norman, OK, 73019, USA

    Qi Cheng & Jincheng Zhuang

  2. Department of Mathematics, Shanghai Jiao Tong University, Shanghai, PR China

    Jiyou Li

Authors
  1. Qi Cheng
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  2. Jiyou Li
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  3. Jincheng Zhuang
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Editor information

Editors and Affiliations

  1. Department of Computer Science and Engineering, Chinese University of Hong Kong, Shatin, 100084, Hong Kong, China

    Leizhen Cai

  2. Department of Computer Science and Engineering, Hong Kong University of Science and Technology, Clear Water Bay, 100044, Hong Kong, China

    Siu-Wing Cheng

  3. Department of Computer Science, University of Hong Kong, Porfkulam, 100084, Hong Kong, China

    Tak-Wah Lam

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Cheng, Q., Li, J., Zhuang, J. (2013). On Determining Deep Holes of Generalized Reed-Solomon Codes. In: Cai, L., Cheng, SW., Lam, TW. (eds) Algorithms and Computation. ISAAC 2013. Lecture Notes in Computer Science, vol 8283. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45030-3_10

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  • DOI: https://doi.org/10.1007/978-3-642-45030-3_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-45029-7

  • Online ISBN: 978-3-642-45030-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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