Optimal Placement Delivery Arrays with Minimum Number of Rows

Coded caching scheme, which is an effective technique to reduce the load during peak traffic times, has recently become quite popular among the coding community. A placement delivery array (PDA in short) can be used to design a coded caching scheme. The number of rows of a PDA corresponds to the subpacketization level in the coded caching scheme. Thus, it is meaningful to construct the optimal PDAs with minimum number of rows. However, no one has yet proved that one of the previously known PDAs (or optimal PDAs) has minimum number of rows. We mainly focus on such optimal PDAs in this paper. We first prove that one class of the optimal PDAs by Maddah-Ali and Niesen has minimum number of rows. Next other two classes of optimal PDAs with minimum number of rows are obtained and proved from a new characterization of a PDA by means of a set of $3$ dimensional vectors and a new derived lower bound.

[1]  Antonia Maria Tulino,et al.  Coded caching with linear subpacketization is possible using Ruzsa-Szeméredi graphs , 2017, 2017 IEEE International Symposium on Information Theory (ISIT).

[2]  Kevin C. Almeroth,et al.  The Use of Multicast Delivery to Provide a Scalable and Interactive Video-on-Demand Service , 1996, IEEE J. Sel. Areas Commun..

[3]  Hooshang Ghasemi,et al.  Improved lower bounds for coded caching , 2015, 2015 IEEE International Symposium on Information Theory (ISIT).

[4]  Jaime Llorca,et al.  Finite-Length Analysis of Caching-Aided Coded Multicasting , 2014, IEEE Transactions on Information Theory.

[5]  Jie Li,et al.  A Framework of Constructions of Minimal Storage Regenerating Codes With the Optimal Access/Update Property , 2013, IEEE Transactions on Information Theory.

[6]  Hooshang Ghasemi,et al.  Improved Lower Bounds for Coded Caching , 2015, IEEE Transactions on Information Theory.

[7]  Deniz Gündüz,et al.  Wireless Content Caching for Small Cell and D2D Networks , 2016, IEEE Journal on Selected Areas in Communications.

[8]  Xiaohu Tang,et al.  Online Coded Caching With Random Access , 2017, IEEE Communications Letters.

[9]  Michael Gastpar,et al.  A new converse bound for coded caching , 2016, 2016 Information Theory and Applications Workshop (ITA).

[10]  Xiaohu Tang,et al.  Optimal Placement Delivery Arrays , 2016, ArXiv.

[11]  Xiaohu Tang,et al.  On the Gap Between Decentralized and Centralized Coded Caching Schemes , 2016, ArXiv.

[12]  Giuseppe Caire,et al.  Fundamental Limits of Caching in Wireless D2D Networks , 2014, IEEE Transactions on Information Theory.

[13]  Xiaohu Tang,et al.  Placement Delivery Array Design Through Strong Edge Coloring of Bipartite Graphs , 2016, IEEE Communications Letters.

[14]  Alexandros G. Dimakis,et al.  Femtocaching and device-to-device collaboration: A new architecture for wireless video distribution , 2012, IEEE Communications Magazine.

[15]  Daniela Tuninetti,et al.  On the optimality of uncoded cache placement , 2015, 2016 IEEE Information Theory Workshop (ITW).

[16]  Mohammad Mohammadi Amiri,et al.  Fundamental Limits of Coded Caching: Improved Delivery Rate-Cache Capacity Trade-off , 2016 .

[17]  Urs Niesen,et al.  Decentralized coded caching attains order-optimal memory-rate tradeoff , 2013, 2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[18]  Jaime Llorca,et al.  Order-Optimal Rate of Caching and Coded Multicasting With Random Demands , 2015, IEEE Transactions on Information Theory.

[19]  Gennian Ge,et al.  Centralized Coded Caching Schemes: A Hypergraph Theoretical Approach , 2016, IEEE Transactions on Information Theory.

[20]  Jehoshua Bruck,et al.  Zigzag Codes: MDS Array Codes With Optimal Rebuilding , 2011, IEEE Transactions on Information Theory.

[21]  Ming Xiao,et al.  Centralized caching in two-layer networks: Algorithms and limits , 2016, 2016 IEEE 12th International Conference on Wireless and Mobile Computing, Networking and Communications (WiMob).

[22]  Urs Niesen,et al.  Online coded caching , 2014, ICC.

[23]  Suhas N. Diggavi,et al.  Hierarchical coded caching , 2014, 2014 IEEE International Symposium on Information Theory.

[24]  T. Charles Clancy,et al.  Improved approximation of storage-rate tradeoff for caching via new outer bounds , 2015, 2015 IEEE International Symposium on Information Theory (ISIT).

[25]  Li Tang,et al.  Coded Caching with Low Subpacketization Levels , 2016, 2016 IEEE Globecom Workshops (GC Wkshps).

[26]  Urs Niesen,et al.  Fundamental Limits of Caching , 2014, IEEE Trans. Inf. Theory.

[27]  Xiaohu Tang,et al.  On the Placement Delivery Array Design for Centralized Coded Caching Scheme , 2015, IEEE Transactions on Information Theory.