Construction of optimal locally recoverable codes and connection with hypergraph

Recently, it was discovered by several authors that a $q$-ary optimal locally recoverable code, i.e., a locally recoverable code archiving the Singleton-type bound, can have length much bigger than $q+1$. This is quite different from the classical $q$-ary MDS codes where it is conjectured that the code length is upper bounded by $q+1$ (or $q+2$ for some special case). This discovery inspired some recent studies on length of an optimal locally recoverable code. It was shown in \cite{LXY} that a $q$-ary optimal locally recoverable code is unbounded for $d=3,4$. Soon after, it was proved that a $q$-ary optimal locally recoverable code with distance $d$ and locality $r$ can have length $\Omega_{d,r}(q^{1 + 1/\lfloor(d-3)/2\rfloor})$. Recently, an explicit construction of $q$-ary optimal locally recoverable codes for distance $d=5,6$ was given in \cite{J18} and \cite{BCGLP}. In this paper, we further investigate construction optimal locally recoverable codes along the line of using parity-check matrices. Inspired by classical Reed-Solomon codes and \cite{J18}, we equip parity-check matrices with the Vandermond structure. It is turns out that a parity-check matrix with the Vandermond structure produces an optimal locally recoverable code must obey certain disjoint property for subsets of $\mathbb{F}_q$. To our surprise, this disjoint condition is equivalent to a well-studied problem in extremal graph theory. With the help of extremal graph theory, we succeed to improve all of the best known results in \cite{GXY} for $d\geq 7$. In addition, for $d=6$, we are able to remove the constraint required in \cite{J18} that $q$ is even.

[1]  Béla Bollobás,et al.  Modern Graph Theory , 2002, Graduate Texts in Mathematics.

[2]  Shlomo Hoory,et al.  The Size of Bipartite Graphs with a Given Girth , 2002, J. Comb. Theory, Ser. B.

[3]  Minghua Chen,et al.  Pyramid Codes: Flexible Schemes to Trade Space for Access Efficiency in Reliable Data Storage Systems , 2007, Sixth IEEE International Symposium on Network Computing and Applications (NCA 2007).

[4]  Enkatesan G Uruswami Unbalanced expanders and randomness extractors from Parvaresh-Vardy codes , 2008 .

[5]  Peter Keevash,et al.  The early evolution of the H-free process , 2009, 0908.0429.

[6]  Cheng Huang,et al.  On the Locality of Codeword Symbols , 2011, IEEE Transactions on Information Theory.

[7]  P. Vijay Kumar,et al.  Optimal linear codes with a local-error-correction property , 2012, 2012 IEEE International Symposium on Information Theory Proceedings.

[8]  Sriram Vishwanath,et al.  Optimal locally repairable codes via rank-metric codes , 2013, 2013 IEEE International Symposium on Information Theory.

[9]  Dimitris S. Papailiopoulos,et al.  Optimal locally repairable codes and connections to matroid theory , 2013, 2013 IEEE International Symposium on Information Theory.

[10]  Itzhak Tamo,et al.  A Family of Optimal Locally Recoverable Codes , 2013, IEEE Transactions on Information Theory.

[11]  Jacques Verstraëte,et al.  A counterexample to sparse removal , 2013, Eur. J. Comb..

[12]  Jacques Verstraëte Extremal problems for cycles in graphs , 2016 .

[13]  Gretchen L. Matthews,et al.  Locally recoverable codes from algebraic curves and surfaces , 2017, ArXiv.

[14]  Chaoping Xing,et al.  Optimal Locally Repairable Codes Via Elliptic Curves , 2017, IEEE Transactions on Information Theory.

[15]  Lingfei Jin,et al.  Explicit Construction of Optimal Locally Recoverable Codes of Distance 5 and 6 via Binary Constant Weight Codes , 2018, IEEE Transactions on Information Theory.

[16]  Yuan Luo,et al.  Optimal Locally Repairable Codes of Distance 3 and 4 via Cyclic Codes , 2019, IEEE Transactions on Information Theory.

[17]  Venkatesan Guruswami,et al.  How Long Can Optimal Locally Repairable Codes Be? , 2019, IEEE Transactions on Information Theory.

[18]  Chaoping Xing,et al.  Construction of Optimal Locally Repairable Codes via Automorphism Groups of Rational Function Fields , 2020, IEEE Transactions on Information Theory.