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Perfect LRCs and k-optimal LRCs

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Abstract

A linear code is called a locally repairable code (LRC) with locality r if one can recover an erased code symbol by accessing at most r other code symbols. Constructions of LRCs have been widely investigated in recent years. In this paper, we give a step forward in this direction. Firstly, we propose a novel concept of perfect LRCs whose size exactly achieves the Hamming-type bound, similar to the perfect codes that achieving the Hamming bound in classical coding theory. By the parity-check matrix approach, we establish some important connections between the existence of LRCs and the existence of some subsets of finite geometry and finite fields with certain properties, respectively. By employing q-Steiner systems and sunflowers in projective geometry and difference sets in finite fields, we obtain two new constructions of perfect LRCs with flexible parameters and present several new constructions of k-optimal LRCs achieving another Hamming-type bound under the integers restriction. Moreover, for fixed q and r, the code lengths of all the q-ary r-LRCs constructed in this paper can be arbitrarily large and the code rates can asymptotically achieve the upper bound \(\frac{r}{r+1}\).

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Acknowledgements

This research is supported in part by National Key R &D Program of China under Grant Nos. 2021YFA1001000 and 2018YFA0704703, the National Natural Science Foundation of China under Grants 62201322, 62171248, 61971243, 12226336 and 12141108, the Natural Science Foundation of Shandong Province under Grant ZR2022QA031, the PCNL KEY project (PCL2021A07), the Guangdong Provincial Key Laboratory of Novel Security Intelligence Technologies (2022B1212010005), the Guangdong Basic and Applied Basic Research Foundation under Grant 2021A1515110066, the GXWD 20220811172936001, the Fundamental Research Funds for the Central Universities, Nankai University, the Natural Science Foundation of Tianjin (20JCZDJC00610). The authors would like to express their sincere gratefulness to the Associate Editor and the two anonymous reviewers for their valuable suggestions and comments which have greatly improved this paper.

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Authors and Affiliations

  1. Key Laboratory of Cryptologic Technology and Information Security, Ministry of Education, Shandong University, Qingdao, 266237, People’s Republic of China

    Weijun Fang

  2. School of Cyber Science and Technology, Shandong University, Qingdao, 266237, People’s Republic of China

    Weijun Fang

  3. Quancheng Laboratory, Jinan, 250103, People’s Republic of China

    Weijun Fang

  4. Department of Computer Science and Technology, Harbin Institute of Technology, Shenzhen, 518055, People’s Republic of China

    Bin Chen

  5. Research Center of Artificial Intelligence, Peng Cheng Laboratory, Shenzhen, 518055, People’s Republic of China

    Bin Chen

  6. Guangdong Provincial Key Laboratory of Novel Security Intelligence Technologies, Shenzhen, 518055, People’s Republic of China

    Bin Chen

  7. Shenzhen International Graduate School, Tsinghua University, Shenzhen, 518055, People’s Republic of China

    Shu-Tao Xia

  8. Chern Institute of Mathematics and LPMC, and Tianjin Key Laboratory of Network and Data Security Technology, Nankai University, Tianjin, 300071, People’s Republic of China

    Fang-Wei Fu

  9. HUAWEI Technologies Co., Ltd., Shenzhen, 518055, People’s Republic of China

    Xiangyu Chen

Authors
  1. Weijun Fang
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  2. Bin Chen
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  3. Shu-Tao Xia
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  4. Fang-Wei Fu
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  5. Xiangyu Chen
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Corresponding author

Correspondence to Bin Chen.

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Communicated by V. A. Zinoviev.

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This paper was presented in part at the 2019 IEEE International Symposium on Information Theory (ISIT) [9], Paris, France and 2020 IEEE International Symposium on Information Theory (ISIT), Los Angeles, California, USA [14].

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Fang, W., Chen, B., Xia, ST. et al. Perfect LRCs and k-optimal LRCs. Des. Codes Cryptogr. 91, 1209–1232 (2023). https://doi.org/10.1007/s10623-022-01148-7

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  • DOI: https://doi.org/10.1007/s10623-022-01148-7

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