Recent years, several new types of codes were introduced to provide fault-tolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes (LRCs for short) have played an important role.A linear code is said to have locality r if each of its code symbols can be repaired by accessing at most r other code symbols. For an LRC with length n, dimension k and locality r, its minimum distance d was proved to satisfy the Singleton-like bound d ≤ n − k − ⌈k/r⌉ + 2. Since then, many works have been done for constructing LRCs meeting the Singleton-like bound over small fields.In this paper, we study quaternary LRCs meeting Singleton-like bound through a parity-check matrix approach. Using tools from finite geometry, we provide some new necessary conditions for LRCs being optimal. From this, we prove that there are 27 different classes of parameters for optimal quaternary LRCs. Moreover, for each class, explicit constructions of corresponding optimal quaternary LRCs are presented.

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Let {(Ai, Bi)} m i=1 be a collection of pairs of sets with |Ai| = a and |Bi| = b for 1 ≤ i ≤ m. Suppose that Ai ∩ Bj = ∅ if and only if i = j, then by the famous Bollobás theorem, we have the size of this collection m ≤ a+b a . In this paper, we consider a variant of this problem by setting {Ai} m i=1 to be intersecting additionally. Using exterior algebra method, we prove a weighted Bollobás type theorem for finite dimensional real vector spaces under these constraints. As a consequence, we have a similar theorem for finite sets, which settles a recent conjecture of Gerbner et. al [10]. Moreover, we also determine the unique extremal structure of {(Ai, Bi)} m i=1 for the primary case of the theorem for finite sets.

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