Repairing Algebraic Geometry Codes

Minimum storage regenerating codes have minimum storage of data in each node and therefore are maximal distance separable (for short) codes. Thus, the number of nodes is upper-bounded by <inline-formula> <tex-math notation="LaTeX">$2^{ {\mathfrak {b}}}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">${\mathfrak {b}}$ </tex-math></inline-formula> is the bits of data stored in each node. From both theoretical and practical points of view (see the details in <xref ref-type="sec" rid="sec1">Section 1</xref>), it is natural to consider regenerating codes that nearly have minimum storage of data, and meanwhile, the number of nodes is unbounded. One of the candidates for such regenerating codes is an algebraic geometry code. In this paper, we generalize the repairing algorithm of Reed–Solomon codes given by Guruswami and Wotters to algebraic geometry codes and present a repairing algorithm for arbitrary one-point algebraic geometry codes. By applying our repairing algorithm to the one-point algebraic geometry codes based on the Garcia–Stichtenoth tower, one can repair a code of rate <inline-formula> <tex-math notation="LaTeX">$1- \varepsilon $ </tex-math></inline-formula> and length <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> over <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula> with bandwidth <inline-formula> <tex-math notation="LaTeX">$(n-1)(1- \tau)\log q$ </tex-math></inline-formula> for any <inline-formula> <tex-math notation="LaTeX">$\varepsilon =2^{(\tau -1/2)\log q}$ </tex-math></inline-formula> with a real <inline-formula> <tex-math notation="LaTeX">$\tau \in (0,1/2)$ </tex-math></inline-formula>. In addition, storage in each node for an algebraic geometry code is close to the minimum storage. Due to nice structures of Hermitian curves, repairing of Hermitian codes is also investigated. As a result, we are able to show that algebraic geometry codes are regenerating codes with good parameters.

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