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Optimal quaternary \((r,\delta )\)-locally recoverable codes: their structures and complete classification

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Abstract

Locally recoverable codes (LRCs) have been introduced as a family of erasure codes that support the repair of a failed storage node by contacting a small number of other nodes in the cluster. Boosted by their applications in distributed storage, LRCs have attracted a lot of attention in recent literature since the concept of codes with locality r was introduced by Gopalan et al. in 2012. Aiming to recover the data from several concurrent node failures, the concept of r-locality was later generalized as \((r, \delta )\)-locality by Prakash et al. An \((r, \delta )\)-LRCs in which every code symbol has \((r, \delta )\)-locality is said to be optimal if it achieves the Singleton-like bound with equality. In present paper, we are interested in optimal \((r, \delta )\)-LRCs over small fields, more precisely, over quaternary field. We study their parity-check matrices or generator matrices, using the properties of projective space. The classification of optimal quaternary \((r,\delta )\)-LRCs and their explicit code constructions are proposed by examining all possible parameters.

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Acknowledgements

The authors are very grateful the Associate Editor and the anonymous referees for their valuable comments and constructive suggestions that improved the quality of this paper highly. The work of Li Xu and Zhengchun Zhou was supported by the NSFC project No. 62071397 and No. 12231015. The work of Jun Zhang was supported by the National Natural Science Foundation of China under the Grants 11971321 and 12222113, and by the National Key Research and Development Program of China under Grant 2018YFA0704703. The last author was supported by the French Agence Nationale de la Recherche through ANR BARRACUDA (ANR-21-CE39-0009).

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Authors and Affiliations

  1. School of Mathematics, Southwest Jiaotong University, Chengdu, 611756, China

    Li Xu & Zhengchun Zhou

  2. School of Mathematical Sciences, Capital Normal University, Beijing, 100048, China

    Jun Zhang

  3. Department of Mathematics, University of Paris VIII, 93526, Saint-Denis, France

    Sihem Mesnager

  4. University of Paris XIII, CNRS, UMR 7539 LAGA, Sorbonne Paris Cité, 93430, Villetaneuse, France

    Sihem Mesnager

  5. Telecom Paris, Polytechnic Institute of Paris, 91120, Palaiseau, France

    Sihem Mesnager

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  1. Li Xu
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  2. Zhengchun Zhou
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  4. Sihem Mesnager
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Correspondence to Sihem Mesnager.

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Communicated by V. A. Zinoviev.

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Xu, L., Zhou , Z., Zhang, J. et al. Optimal quaternary \((r,\delta )\)-locally recoverable codes: their structures and complete classification. Des. Codes Cryptogr. 91, 1495–1526 (2023). https://doi.org/10.1007/s10623-022-01165-6

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