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As an important coding scheme in modern distributed storage systems, locally repairable codes (LRCs) have attracted a lot of attentions from perspectives of both practical applications and theoretical research. As a major topic in the research of LRCs, bounds and constructions of the corresponding optimal codes are of particular concerns. In this work, codes with $(r,\delta)$ -locality which have optimal minimal distance w.r.t. the bound given by Prakash et al. are considered. Through parity-check matrix approach, constructions of both optimal $(r,\delta)$ -LRCs with all symbol locality ($(r,\delta)_{a}$ -LRCs) and optimal $(r,\delta)$ -LRCs with information locality ($(r,\delta)_{i}$ -LRCs) are provided. As a generalization of a work of Xing and Yuan, these constructions are built on a connection between sparse hypergraphs and optimal $(r,\delta)$ -LRCs. With the help of constructions of large sparse hypergraphs, the lengths of codes obtained from our construction can be super-linear in the alphabet size. This improves upon previous constructions when the minimal distance of the code is at least $3\delta +1$. As two applications, optimal H-LRCs with super-linear lengths and GSD codes with unbounded lengths are also constructed. [ABSTRACT FROM AUTHOR]

Given a database, the private information retrieval (PIR) protocol allows a user to make queries to several servers and retrieve a certain item of the database via the feedbacks without revealing the identity of the specific item to any single server. Classic $k$ -server PIR protocols work on replicated databases, i.e., each of the $k$ servers stores a whole copy of the database. Recently, new PIR models were proposed with coding techniques arising from the distributed storage system. In these new models, each server only stores a fraction $1/s$ of the whole database, where $s>1$ is the given rational number. The PIR array codes are recently proposed by Fazeli, Vardy, and Yaakobi to characterize the new models. The central problem in designing a PIR array code with $m$ servers and the $k$ -PIR property (which indicates that these $m$ servers may emulate a classic $k$ -server PIR protocol) is to maximize $k/m$ , known as the virtual server rate. Our main contribution to this problem is twofold. First, for the case $1 < s\le 2$ , although the PIR array codes with optimal rate have been constructed recently by Blackburn and Etzion, the number of servers in their construction is rather large. We determine the minimum number of servers admitting the existence of a PIR array code with an optimal rate for a certain range of parameters. Second, for the case $s>2$ , a new upper bound on the rate of a PIR array code is presented. Besides, we also have some discussions on an asymptotically optimal construction by Blackburn and Etzion. [ABSTRACT FROM AUTHOR]

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 . [ABSTRACT FROM PUBLISHER]

Various kinds of secure codes and their related hash families are broadly studied combinatorial structures for protecting copyrighted materials. The codewords in such a structure can be regarded as a subset of Q^N , the set of all $q$ -ary vectors of given length $N$ , satisfying some constraints. We use a hypergraph model to characterize the combinatorial structure. By applying a result of Duke et al. on the lower bound of the independence number of hypergraphs, we provide a new approach to evaluate the lower bounds for several kinds of secure codes and related hash families. In particular, the general method is illustrated via the examples of existence results on some perfect hash families, frameproof codes, and separable codes. [ABSTRACT FROM PUBLISHER]

Multiply constant-weight codes (MCWCs) were introduced recently to improve the reliability of certain physically unclonable function response. In this paper, the bounds of MCWCs and the constructions of optimal MCWCs are studied. First, we derive three different types of upper bounds which improve the Johnson-type bounds given by Chee et al. for some parameters. The asymptotic lower bound of MCWCs is also examined. Then, we obtain the asymptotic existence of two classes of optimal MCWCs, which shows that the Johnson-type bounds for MCWCs with distances 2\sum i=1^{m}wi-2 or $2mw-2w$ are asymptotically exact. Finally, we construct a class of optimal MCWCs with total weight four and distance six by establishing the connection between such MCWCs and a new kind of combinatorial structures. As a consequence, the maximum sizes of MCWCs with total weight less than or equal to four are determined almost completely. [ABSTRACT FROM AUTHOR]

Multimedia fingerprinting is an effective technique to trace the sources of pirate copies of copyrighted multimedia contents. Separable codes were introduced to detect colluders taking part in the averaging attack, which is the most feasible approach to perform a collusion attack. In this paper, we provide some improved bounds for the size of separable codes. Using a combinatorial technique named grouping coordinates, we have greatly reduced the upper bound. On the other hand, we also provide some lower bounds for the size of separable codes by both probabilistic and deterministic construction methods. In particular, \( \overline {2}\) -separable codes with asymptotically optimal rate are obtained by the deletion method. [ABSTRACT FROM AUTHOR]

Constant-weight codes (CWCs) play an important role in coding theory. The problem of determining the sizes for optimal ternary CWCs with length n, weight 4, and minimum Hamming distance 5 ((n,5,4)3 code) has been settled for all positive integers n\leq 10 or n > 10 and n\equiv 1\pmod 3 with n\in \13,52,58\ undetermined. In this paper, we investigate the problem of constructing optimal (n,5,4)3 codes for all lengths n with the tool of group divisible codes. We determine the size of an optimal (n,5,4)3 code for each integer n\geq 4 leaving the lengths n\in \12,13,21,27,33,39,45,52\ unsolved. [ABSTRACT FROM PUBLISHER]