We consider constructing capacity-achieving linear codes with minimum message size for private information retrieval (PIR) from N non-colluding databases, where each message is coded using maximum distance separable (MDS) codes, such that it can be recovered from accessing the contents of any T databases. It is shown that the minimum message size (sometimes also referred to as the sub-packetization factor) is significantly, in fact exponentially, lower than previously believed. More precisely, when K > T / gcd(N, T ) where K is the total number of messages in the system and gcd(·, ·) means the greatest common divisor, we establish, by providing both novel code constructions and a matching converse, the minimum message size as lcm(N − T, T ), where lcm(·, ·) means the least common multiple. On the other hand, when K is small, we show that it is in fact possible to design codes with a message size even smaller than lcm(N − T, T ). arXiv:1903.08229v2 [cs.IT] 22 Jan 2020 significantly restricts the possible usage scenarios. Moreover, a large message size also usually implies that the encoding and the decoding functions are more complex, which not only requires more engineering efforts to implement but also hinders the efficiency of the system operation. From a theoretical point of view, a code with a smaller message size usually implies a more transparent coding structure, which can be valuable for related problems; see, e.g., [7] for such an example. Thus codes with a smaller message size are highly desirable in both theory and practice.The capacity-achieving code given in [4] requires L = T N K , which can be extremely large for a system with even a moderate number of messages. The problem of reducing the message size of capacity-achieving codes was recently considered by Xu and Zhang [6], and it was shown that under the assumption that all answers are of the same length, the message size must satisfy L ≥ T (N/ gcd(N, T )) K−1 . These existing results may have left the impression that capacityachieving codes would necessitate a message size exponential in the number of messages.In this work, we show that the minimum message size for capacity-achieving PIR codes can in fact be significantly smaller than previously believed, by providing capacity-achieving linear codes with message size L = lcm(N −T, T ). Two linear code constructions, referred to as Construction-A and Construction-B, respectively, are given. The two constructions have the same download cost and message size, however Construction-B has a better upload cost (i.e., a lower communication cost for the user to send the queries), at the expense of being slightly more sophisticated than Construction-A. The key difference between the two proposed constructions and existing codes in the literature is that the proposed codes reduce the reliance on the so-called variety symmetry [8], which should be distinguished from the asymmetry discussed in [9], and the answers may be of different lengths 1 . We further show that this is in fact the minimum message size whe...

Motivated by the engineering problem of cellular coverage optimization, we propose a novel multiarmed bandit model called generalized global bandit. We develop a series of greedy algorithms that have the capability to handle nonmonotonic but decomposable reward functions, multidimensional global parameters, and switching costs. The proposed algorithms are rigorously analyzed under the multiarmed bandit framework, where we show that they achieve bounded regret, and hence, they are guaranteed to converge to the optimal arm in finite time. The algorithms are then applied to the cellular coverage optimization problem to achieve the optimal tradeoff between sufficient small cell coverage and limited macroleakage without prior knowledge of the deployment environment. The performance advantage of the new algorithms over existing bandits solutions is revealed analytically and further confirmed via numerical simulations. The key element behind the performance improvement is a more efficient "trial and error" mechanism, in which any trial will help improve the knowledge of all candidate power levels.

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