Locally recoverable codes with availability t≥2 from fiber products of curves

Kathryn Haymaker; Beth Malmskog; Gretchen L. Matthews

doi:10.3934/amc.2018020

Article Contents

2018, Volume 12, Issue 2: 317-336. Doi: 10.3934/amc.2018020

This issue Previous Article Several infinite families of p-ary weakly regular bent functions Next Article Completely regular codes by concatenating Hamming codes

Locally recoverable codes with availability t≥2 from fiber products of curves

1.
Department of Mathematics and Statistics, Villanova University, Villanova, PA 19085, USA
2.
Department of Mathematics and Computer Science, Colorado College, Colorado Springs, CO 80903, USA
3.
Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, USA

* Corresponding author: Beth Malmskog
* Corresponding author: Beth Malmskog

Received: November 30, 2016

Revised: August 31, 2017

Published: April 2018

The second author is supported by NSA grant H98230-16-1-0300.

Abstract Full Text(HTML) Figure(6) / Table(1) Related Papers Cited by

Abstract

We generalize the construction of locally recoverable codes on algebraic curves given by Barg, Tamo and Vlăduţ [4] to those with arbitrarily many recovery sets by exploiting the structure of fiber products of curves. Employing maximal curves, we create several new families of locally recoverable codes with multiple recovery sets, including codes with two recovery sets from the generalized Giulietti and Korchmáros (GK) curves and the Suzuki curves, and new locally recoverable codes with many recovery sets based on the Hermitian curve, using a fiber product construction of van der Geer and van der Vlugt. In addition, we consider the relationship between local error recovery and global error correction as well as the availability required to locally recover any pattern of a fixed number of erasures.

Keywords:

Mathematics Subject Classification: Primary: 14G50, 94B27; Secondary: 11T71.

Citation:

Full Text(HTML)

Figure 1. A visualization of points on a fiber product of two curves. Points on the fiber product $\mathcal{X}$ may be thought of as tuples of points on the curves $\mathcal{Y}_1$ and $\mathcal{Y}_2$ which lie above the same point on $\mathcal{Y}$.

Download: Full-size image PowerPoint slide

Figure 2. The fiber product $\mathcal{X}$ of $t$ curves $\mathcal{Y}_j$.

Download: Full-size image PowerPoint slide

Figure 3. Function fields associated with the fiber product.

Download: Full-size image PowerPoint slide

Figure 4. Generalized GK curve as a fiber product.

Download: Full-size image PowerPoint slide

Figure 5. Suzuki curve and its quotients used for constructing LRC(2) with balanced recovery sets.

Download: Full-size image PowerPoint slide

Figure 6. Curves for locally recoverable codes with availability $t$.

Download: Full-size image PowerPoint slide

Table 1. The generalized GK curves $\mathcal{C}_3$ over $\mathbb{F}_{729}$ produce LRC(2)s of length $n = 6048$, with $N = 3$, $q = 3$, $r_1 = 6$, $r_2 = 2$, and $D = l\infty_y$, with $l$ determining $k$ and $d$ as above.

$l$	$k$	$d\geq$
270	3252	215
260	3132	425
250	3012	635
240	2892	845
230	2772	1055
220	2652	1265
210	2532	1475

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

	M. Abdón , J. Bezerra and L. Quoos , Further examples of maximal curves, Journal of Pure and Applied Algebra, 213 (2009) , 1192-1196. doi: 10.1016/j.jpaa.2008.11.037.
	E. Ballico and A. Ravagnani , Embedding Suzuki curves in $\Bbb P^4$, Journal of Commutative Algebra, 7 (2015) , 149-166. doi: 10.1216/JCA-2015-7-2-149.
	A. Barg, K. Haymaker, E. W. Howe, G. L. Matthews and A. Várilly-Alvarado, Locally recoverable codes from algebraic curves and surfaces, Algebraic Geometry for Coding Theory and Cryptography, (2017), 95–127, arXiv: 1701.05212. doi: 10.1007/978-3-319-63931-4_4.
	A. Barg, I. Tamo and S. Vlădut¸, Locally recoverable codes on algebraic curves, Proceedings of the IEEE Int. Symp. Info. Theory, (2015), 1252–1256, Extended version: arXiv: 1603.08876. doi: 10.1109/ISIT.2015.7282656.
	A. Eid and I. Duursma, Smooth embeddings for the Suzuki and Ree curves, Proceedings of the conference on Arithmetic, Geometry and Coding Theory (AGCT 2013), Contemporary Mathematics Series (AMS), 637 (2015), 251–291.
	A. Garcia , C. Güneri and H. Stichtenoth , A generalization of the Giulietti-Korchmáros maximal curve, Advances in Geometry, 10 (2010) , 427-434.
	M. Giulietti and G. Korchmáros , A new family of maximal curves over a finite field, Math. Ann., 343 (2009) , 229-245. doi: 10.1007/s00208-008-0270-z.
	M. Giulietti , G. Korchmáros and F. Torres , Quotient curves of the Suzuki curve, Acta Arithmetica, 122 (2006) , 245-274. doi: 10.4064/aa122-3-3.
	R. Guralnick , B. Malmskog and R. Pries , The automorphism groups of a family of maximal curves, Journal of Algebra, 361 (2012) , 92-106. doi: 10.1016/j.jalgebra.2012.03.036.
	J. Hansen, Deligne-Lusztig varieties and group codes, in Coding Theory and Algebraic Geometry, Lecture Notes in Mathematics, 1518 (1992), 63–81.
	Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics, 2002.
	H.-G. Rück and H. Stichtenoth , A characterization of Hermitian function fields over finite fields, J. Reine Angew. Math., 457 (1994) , 185-188.
	H. Stichtenoth, Algebraic Function Fields and Codes, Springer-Verlag, Berlin, 2009.
	G. van der Geer and M. van der Vlugt, How to construct curves over finite fields with many points, in Arithmetic Geometry (Cortona, 1994), Symposia Mathematica Cambridge: Cambridge University Press, 37 (1997), 169–189.