Construction of optimal locally recoverable codes and connection with hypergraph
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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 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 Ω_d,r(q^1 + 1/(d-3)/2). Recently, an explicit construction of q-ary optimal locally recoverable codes for distance d=5,6 was given in J18 and 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 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 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 GXY for d≥ 7. In addition, for d=6, we are able to remove the constraint required in J18 that q is even.
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