A locally recoverable code with locality $ \bf r $ can recover the missing coordinate from at most $ {\bf r} $ symbols. The locally recoverable codes have attracted a lot of attention because they are more advanced coding techniques that are applied to distributed and cloud storage systems. In this work, we focus on locally recoverable codes in Hermitian function fields over $ \Bbb F_{q^2} $, where $ q $ is a prime power. With a certain type of divisor, we obtain an improved lower bound of the minimum distance for locally recoverable codes in Hermitian function fields. For doing this, we give explicit formulae of the dimension for some divisors of Hermitian function fields. We also present a standard that tells us when a divisor with certain places suggests an improved lower bound.
Citation: Boran Kim. Locally recoverable codes in Hermitian function fields with certain types of divisors[J]. AIMS Mathematics, 2022, 7(6): 9656-9667. doi: 10.3934/math.2022537
A locally recoverable code with locality $ \bf r $ can recover the missing coordinate from at most $ {\bf r} $ symbols. The locally recoverable codes have attracted a lot of attention because they are more advanced coding techniques that are applied to distributed and cloud storage systems. In this work, we focus on locally recoverable codes in Hermitian function fields over $ \Bbb F_{q^2} $, where $ q $ is a prime power. With a certain type of divisor, we obtain an improved lower bound of the minimum distance for locally recoverable codes in Hermitian function fields. For doing this, we give explicit formulae of the dimension for some divisors of Hermitian function fields. We also present a standard that tells us when a divisor with certain places suggests an improved lower bound.
[1] | E. Ballico, C. Marcolla, Higher hamming weights for locally recoverable codes on algebraic curves, Finite Fields Th. App., 40 (2016), 61–72. https://doi.org/10.1016/j.ffa.2016.03.004 doi: 10.1016/j.ffa.2016.03.004 |
[2] | A. Barg, I. Tamo, S. Vlǎduţ, Locally recoverable codes on algebraic curves, IEEE T. Inform. Theory, 63 (2017), 4928–4939. https://doi.org/10.1109/TIT.2017.2700859 doi: 10.1109/TIT.2017.2700859 |
[3] | P. Gopalan, C. Huang, H. Simitci, S. Yekhanin, On the locality of codeword symbols, IEEE T. Inform. theory, 58 (2012), 6925–6934. https://doi.org/10.1109/TIT.2012.2208937 doi: 10.1109/TIT.2012.2208937 |
[4] | L. Jin, H. Kan, Y. Zhang, Constructions of locally repairable codes with multiple recovering sets via rational function fields. IEEE T. Inform. Theory, 66 (2019), 202–209. https://doi.org/10.1109/TIT.2019.2946627 |
[5] | S. Kruglik, K. Nazirkhanova, A. Frolov, New bounds and generalizations of locally recoverable codes with availability, IEEE T. Inform. Theory, 65 (2019), 4156–4166. https://doi.org/10.1109/TIT.2019.2897705. doi: 10.1109/TIT.2019.2897705 |
[6] | X. Li, L. Ma, C. Xing, Optimal locally repairable codes via elliptic curves, IEEE T. Inform. Theory, 65 (2018), 108–117. https://doi.org/10.1109/TIT.2018.2844216 doi: 10.1109/TIT.2018.2844216 |
[7] | Y. Luo, C. Xing, C. Yuan, Optimal locally repairable codes of distance 3 and 4 via cyclic codes, IEEE T. Inform. Theory, 65 (2018), 1048–1053. https://doi.org/10.1109/TIT.2018.2854717 doi: 10.1109/TIT.2018.2854717 |
[8] | H. Maharaj, G. L. Matthews, G. Pirsic, Riemann-roch spaces of the hermitian function field with applications to algebraic geometry codes and low-discrepancy sequences, J. Pure Appl. Algebra, 195 (2005), 261–280. https://doi.org/10.1016/j.jpaa.2004.06.010 doi: 10.1016/j.jpaa.2004.06.010 |
[9] | C. Munuera, W. Tenório, F. Torres, Locally recoverable codes from algebraic curves with separated variables, Adv. Math. Commun., 14 (2020), 265. https://doi.org/10.1587/bplus.14.265 doi: 10.1587/bplus.14.265 |
[10] | I. Tamo, A. Barg, Bounds on locally recoverable codes with multiple recovering sets, In: 2014 IEEE International Symposium on Information Theory, 2014,691–695. |
[11] | I. Tamo, A. Barg, A family of optimal locally recoverable codes, IEEE T. Inform. Theory, 60 (2014), 4661–4676. https://doi.org/10.1109/TIT.2014.2321280 doi: 10.1109/TIT.2014.2321280 |
[12] | A. Wang, Z. Zhang, Repair locality with multiple erasure tolerance, IEEE T. Inform. Theory, 60 (2014), 6979–6987. https://doi.org/10.1109/TIT.2014.2351404 doi: 10.1109/TIT.2014.2351404 |