Abstract
According to a magnific method due to I. Tamo and A. Barg, a class of polynomials over finite fields, called good polynomials, was introduced and used to construct optimal Locally Recoverable Codes (LRC), which have been developed and exploited in distributed storage. An important derived algebraic problem is, for a given finite field \(\mathbb {F}_q\) and a fixed integer r, to find a polynomial of degree \(r+1\) that is constant on as many subsets of \(\mathbb {F}_q\) as possible of size \(r+1\). Compared to the literature on this topic, our main contribution is introducing a new parameter that measures how “good” a polynomial is in the sense of LRC. Our new approach allows us to characterize completely good polynomials of a low degree over finite fields and, next, to derive new constructions of such polynomials, leading to optimal LRC with new flexible localities. Specifically, several good polynomials of degree at most 6 are studied and described precisely in this paper.
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13 July 2021
A Correction to this paper has been published: https://doi.org/10.1007/s10623-021-00909-0
References
Dickson L.E.: Criteria for the irreducibility of functions in a finite field. Bull. Am. Math. Soc. 13(1), 1–8 (1906).
Ding C.: Codes from Difference Sets. World Scientific, Singapore (2014).
Gopalan P., Huang C., Simitci H., Yekhanin S.: On the locality of codeword symbols. IEEE Trans. Inf. Theory 58(11), 6925–6934 (2012).
Goparaju S., Calderbank A.R.: Binary cyclic codes that are locally repairable. pp. 676–680 (2014).
Kyureghyan G., Li S., Pott A.: On the intersection distribution of degree three polynomials and related topics. arXiv preprint arXiv:2003.10040 (2020).
Kyureghyan G., Müller P., Wang Q.: On the size of kakeya sets in finite vector spaces. arXiv preprint arXiv:1302.5591 (2013).
Leonard P.A.: On factoring quartics (mod p). J. Number Theory 1(1), 113–115 (1969).
Li S., Pott A.: Intersection distribution, non-hitting index and kakeya sets in affine planes. Finite Fields Appl. 66, 101691 (2020).
Li X., Ma L., Xing C.: Optimal locally repairable codes via elliptic curves. IEEE Trans. Inf. Theory 65(1), 108–117 (2019).
Liu J., Mesnager S., Chen L.: New constructions of optimal locally recoverable codes via good polynomials. IEEE Trans. Inf. Theory 64(2), 889–899 (2018).
Liu J., Mesnager S., Tang D.: Constructions of optimal locally recoverable codes via dickson polynomials. Des. Codes Cryptogr. To appear (2020).
Luo Y., Xing C., Yuan C.: Optimal locally repairable codes of distance 3 and 4 via cyclic codes. IEEE Trans. Inf. Theory 65(2), 1048–1053 (2019).
Micheli G.: Constructions of locally recoverable codes which are optimal. IEEE Trans. Inf. Theory 66(1), 167–175 (2020).
Papailiopoulos D.S., Dimakis A.G.: Locally repairable codes. In: 2012 IEEE International Symposium on Information Theory Proceedings, pp. 2771–2775 (2012).
Silberstein N., Rawat A.S., Koyluoglu O.O., Vishwanath S.: Optimal locally repairable codes via rank-metric codes. In: 2013 IEEE International Symposium on Information Theory Proceedings, pp. 1819–1823 (2013).
Silberstein N., Zeh A.: Optimal binary locally repairable codes via anticodes. In: 2015 IEEE International Symposium on Information Theory Proceedings, pp. 1247–1251 (2015).
Song W., Dau S.H., Yuen C., Li T.J.: Optimal locally repairable linear codes. IEEE J. Sel. Areas Commun. 32(5), 1019–1036 (2014).
Tamo I., Barg A.: A family of optimal locally recoverable codes. IEEE Trans. Inf. Theory 60(8), 4661–4676 (2014).
Tamo I., Papailiopoulos D.S., Dimakis A.G.: Optimal locally repairable codes and connections to matroid theory. In: 2013 IEEE International Symposium on Information Theory Proceedings, pp. 1814–1818 (2013).
Williams K.S.: Note on cubics over \({\rm GF}(2^n)\) and \({\rm GF}(3^n)\). J. Number Theory 7(4), 361–365 (1975).
Zeh A., Yaakobi E.: Optimal linear and cyclic locally repairable codes over small fields. In: 2015 IEEE Information Theory Workshop (ITW), pp. 1–5 (2015).
Acknowledgements
The authors thank the Assoc. Edit. and the anonymous reviewers for their valuable comments, which have highly improved the quality and the presentation of the paper. The work of Chang-An Zhao is partially supported by National Key R&D Program of China under Grant No. 2017YFB0802500, by NSFC under Grant No. 61972428, by the Major Program of Guangdong Basic and Applied Research under Grant No. 2019B030302008 and by the Open Fund of State Key Laboratory of Information Security (Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093) under Grant No. 2020-ZD-02.
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Chen, R., Mesnager, S. & Zhao, CA. Good polynomials for optimal LRC of low locality. Des. Codes Cryptogr. 89, 1639–1660 (2021). https://doi.org/10.1007/s10623-021-00886-4
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DOI: https://doi.org/10.1007/s10623-021-00886-4