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We investigate the problem of coded caching for nonuniform demands when the structured clique cover algorithm proposed by Maddah-Ali and Niesen for decentralized caching is used for delivery. We apply this algorithm to all user demands regardless of their request probabilities. This allows for coding among the files that have different request probabilities but makes the allocation of memory to different files challenging during the content placement phase. As our main contribution, we analytically characterize the optimal placement strategy that minimizes the expected delivery rate under a storage capacity constraint. It is shown that the optimal placement follows either a two or a three group strategy, where a set of less popular files are not cached at all and the files within each of the other sets are allocated identical amounts of storage as if they had the same request probabilities. We show that for a finite set of storage capacities, that we call the base-cases of the problem, the two group strategy is always optimal. For other storage capacities, optimal placement is achieved by memory sharing between certain base-cases and the resulting placement either follows a two or a three group strategy depending on the corresponding base-cases used. We derive a polynomial time algorithm that determines the base-cases of the problem given the number of caches and popularity distribution of files. Given the base-cases of the problem, the optimal memory allocation parameters for any storage capacity are derived analytically., Comment: Submitted to IEEE Transactions on Information Theory

Coded caching is an effective technique to reduce the redundant traffic in wireless networks. The existing coded caching schemes require the splitting of files into a possibly large number of subfiles, i.e., they perform coded subfile caching. Keeping the files intact during the caching process would actually be appealing, broadly speaking because of its simpler implementation. However, little is known about the effectiveness of this coded file caching in reducing the data delivery rate. In this paper, we propose such a file caching scheme which uses a decentralized algorithm for content placement and either an online clique cover or matching algorithm for the delivery of missing data. We derive approximations for the expected delivery rate of both schemes using the differential equations method, and show them to be tight through concentration analysis and computer simulations. Our numerical results demonstrate that the proposed coded file caching is significantly more effective than uncoded caching in reducing the delivery rate. We furthermore show the additional improvement in the performance of the proposed scheme when its application is extended to subfile caching with a small number of subfiles., Comment: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=8614426&isnumber=4656680

The problem of content delivery in caching networks is investigated for scenarios where multiple users request identical files. Redundant user demands are likely when the file popularity distribution is highly non-uniform or the user demands are positively correlated. An adaptive method is proposed for the delivery of redundant demands in caching networks. Based on the redundancy pattern in the current demand vector, the proposed method decides between the transmission of uncoded messages or the coded messages of [1] for delivery. Moreover, a lower bound on the delivery rate of redundant requests is derived based on a cutset bound argument. The performance of the adaptive method is investigated through numerical examples of the delivery rate of several specific demand vectors as well as the average delivery rate of a caching network with correlated requests. The adaptive method is shown to considerably reduce the gap between the non-adaptive delivery rate and the lower bound. In some specific cases, using the adaptive method, this gap shrinks by almost 50% for the average rate., Comment: 8 pages,8 figures. Submitted to IEEE transaction on Communications in 2015. A short version of this article was published as an IEEE Communications Letter with DOI: 10.1109/LCOMM.2016.2558144

A number of recent works have used a variety of combinatorial constructions to derive Tanner graphs for LDPC codes and some of these have been shown to perform well in terms of their probability of error curves and error floors. Such graphs are bipartite and many of these constructions yield biregular graphs where the degree of left vertices is a constant $c+1$ and that of the right vertices is a constant $d+1$. Such graphs are termed $(c+1,d+1)$ biregular bipartite graphs here. One property of interest in such work is the girth of the graph and the number of short cycles in the graph, cycles of length either the girth or slightly larger. Such numbers have been shown to be related to the error floor of the probability of error curve of the related LDPC code. Using known results of graph theory, it is shown how the girth and the number of cycles of length equal to the girth may be computed for these $(c+1,d+1)$ biregular bipartite graphs knowing only the parameters $c$ and $d$ and the numbers of left and right vertices. While numerous algorithms to determine the number of short cycles in arbitrary graphs exist, the reduction of the problem from an algorithm to a computation for these biregular bipartite graphs is of interest., Comment: One of the results has been shown invalid via counterexample

The use of partial geometries to construct parity-check matrices for LDPC codes has resulted in the design of successful codes with a probability of error close to the Shannon capacity at bit error rates down to $10^{-15}$. Such considerations have motivated this further investigation. A new and simple construction of a type of partial geometries with quasi-cyclic structure is given and their properties are investigated. The trapping sets of the partial geometry codes were considered previously using the geometric aspects of the underlying structure to derive information on the size of allowable trapping sets. This topic is further considered here. Finally, there is a natural relationship between partial geometries and strongly regular graphs. The eigenvalues of the adjacency matrices of such graphs are well known and it is of interest to determine if any of the Tanner graphs derived from the partial geometries are good expanders for certain parameter sets, since it can be argued that codes with good geometric and expansion properties might perform well under message-passing decoding., Comment: 34 pages with single column, 6 figures

Aspects of the properties, enumeration and construction of points on diagonal and Hermitian surfaces have been considered extensively in the literature and are further considered here. The zeta function of diagonal surfaces is given as a direct result of the work of Wolfmann. Recursive construction techniques for the set of rational points of Hermitian surfaces are of interest. The relationship of these techniques here to the construction of codes on surfaces is briefly noted.