In this paper, locally repairable codes which have optimal minimum Hamming distance with respect to the bound presented by Prakash et al. are considered. New upper bounds on the length of such optimal codes are derived. The new bounds apply to more general cases, and have weaker requirements compared with the known ones. In this sense, they both improve and generalize previously known bounds. Further, optimal codes are constructed, whose length is order-optimal with respect to the new upper bounds. Notably, the length of the codes is super-linear in the alphabet size.

We construct maximally recoverable codes (corresponding to partial MDS codes) which are based on linearized Reed-Solomon codes. The new codes have a smaller field size requirement compared with known constructions. For certain asymptotic regimes, the constructed codes have order-optimal alphabet size, asymptotically matching the known lower bound.

Locally repairable codes are desirable for distributed storage systems to improve the repair efficiency. In this paper, a new combination of codes with locality and codes with multiple disjoint repair sets (also called availability) is introduced. Accordingly, a Singleton-type bound is derived for the new code, which contains those bounds in [9], [20], [28] as special cases. Optimal constructions are proposed with respect to the new bound. In addition, these constructions can also generate optimal codes with multiple disjoint repair sets with respect to the bound in [28], which to the best of our knowledge, are the first explicit constructions that can achieve the bound in [28].

Frameproof codes are used to fingerprint digital data. They can prevent copyrighted materials from unauthorized use. In this paper, we study upper and lower bounds for $w$ -frameproof codes of length $N$ over an alphabet of size $q$ . The upper bound is based on a combinatorial approach and the lower bound is based on a probabilistic construction. Both bounds can improve one of the previous results when $q$ is small compared with $w$ , say $cq\leq w$ for some constant $c\leq q$ . Furthermore, we pay special attention to binary frameproof codes. We show a binary $w$ -frameproof code of length $N$ cannot have more than $N$ codewords if $N<\binom {w+1}{2}$ .

Multimedia fingerprinting is an effective technique to trace the sources of pirate copies of copyrighted multimedia information. Separable codes can be used to construct fingerprints resistant to the averaging collusion attack on multimedia contents. In this paper, we investigate -separable codes from a combinatorial point of view. We first derive several upper bounds on the sizes of -separable codes, and then turn our attention to the constructions of optimal -separable codes with short length. Two infinite families of optimal -separable codes of length 2 are constructed from projective planes, and all optimal -separable codes of length 3 are explicitly constructed by means of difference matrices. These optimal -separable codes with short length can be used to construct good -separable codes with long length by a known composition construction.