Constructions of Maximally Recoverable Local Reconstruction Codes via Function Fields

Authors Venkatesan Guruswami, Lingfei Jin , Chaoping Xing

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Author Details

Venkatesan Guruswami
  • Computer Science Department, Carnegie Mellon University, Pittsburgh, PA, USA
Lingfei Jin
  • Shanghai Key Laboratory of Intelligent Information Processing, School of Computer Science, Fudan University, Shanghai, China
  • Shanghai Institute of Intelligent Electronics & Systems, Shanghai, China
  • Shanghai Bolckchain Engineering Research Center, Fudan University, Shanghai 200433, China
Chaoping Xing
  • School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore

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Venkatesan Guruswami, Lingfei Jin, and Chaoping Xing. Constructions of Maximally Recoverable Local Reconstruction Codes via Function Fields. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 68:1-68:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


Local Reconstruction Codes (LRCs) allow for recovery from a small number of erasures in a local manner based on just a few other codeword symbols. They have emerged as the codes of choice for large scale distributed storage systems due to the very efficient repair of failed storage nodes in the typical scenario of a single or few nodes failing, while also offering fault tolerance against worst-case scenarios with more erasures. A maximally recoverable (MR) LRC offers the best possible blend of such local and global fault tolerance, guaranteeing recovery from all erasure patterns which are information-theoretically correctable given the presence of local recovery groups. In an (n,r,h,a)-LRC, the n codeword symbols are partitioned into r disjoint groups each of which include a local parity checks capable of locally correcting a erasures. The codeword symbols further obey h heavy (global) parity checks. Such a code is maximally recoverable if it can correct all patterns of a erasures per local group plus up to h additional erasures anywhere in the codeword. This property amounts to linear independence of all such subsets of columns of the parity check matrix. MR LRCs have received much attention recently, with many explicit constructions covering different regimes of parameters. Unfortunately, all known constructions require a large field size that is exponential in h or a, and it is of interest to obtain MR LRCs of minimal possible field size. In this work, we develop an approach based on function fields to construct MR LRCs. Our method recovers, and in most parameter regimes improves, the field size of previous approaches. For instance, for the case of small r << epsilon log n and large h >=slant Omega(n^{1-epsilon}), we improve the field size from roughly n^h to n^{epsilon h}. For the case of a=1 (one local parity check), we improve the field size quadratically from r^{h(h+1)} to r^{h floor[(h+1)/2]} for some range of r. The improvements are modest, but more importantly are obtained in a unified manner via a promising new idea.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Coding theory
  • Erasure codes
  • Algebraic constructions
  • Linear algebra
  • Locally Repairable Codes
  • Explicit constructions


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