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

  • * Corresponding author: Beth Malmskog

    * Corresponding author: Beth Malmskog 

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

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  • 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.

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

    Citation:

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  • 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 $\mathcal{Y}$.

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

    Figure 3.  Function fields associated with the fiber product.

    Figure 4.  Generalized GK curve as a fiber product.

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

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

    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
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  •   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.
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