1,822 research outputs found
Divisible Design Graphs
AMS Subject Classification: 05B05, 05E30, 05C50.Strongly regular graph;Group divisible design;Deza graph;(v;k;)-Graph
Coded Caching based on Combinatorial Designs
We consider the standard broadcast setup with a single server broadcasting
information to a number of clients, each of which contains local storage
(called \textit{cache}) of some size, which can store some parts of the
available files at the server. The centralized coded caching framework,
consists of a caching phase and a delivery phase, both of which are carefully
designed in order to use the cache and the channel together optimally. In prior
literature, various combinatorial structures have been used to construct coded
caching schemes. In this work, we propose a binary matrix model to construct
the coded caching scheme. The ones in such a \textit{caching matrix} indicate
uncached subfiles at the users. Identity submatrices of the caching matrix
represent transmissions in the delivery phase. Using this model, we then
propose several novel constructions for coded caching based on the various
types of combinatorial designs. While most of the schemes constructed in this
work (based on existing designs) have a high cache requirement (uncached
fraction being or , being
the number of users), they provide a rate that is either constant or decreasing
() with increasing , and moreover require competitively
small levels of subpacketization (being ), which is an
extremely important parameter in practical applications of coded caching. We
mark this work as another attempt to exploit the well-developed theory of
combinatorial designs for the problem of constructing caching schemes,
utilizing the binary caching model we develop.Comment: 10 pages, Appeared in Proceedings of IEEE ISIT 201
Combinatorial Designs with Two Singular Values II. Partial Geometric Designs
In this and an earlier paper [17] we study combinatorial designs whose incidence matrix has two distinct singular values.These generalize (v, k, Γ«) designs, and include uniform multiplicative designs and partial geometric designs.Here we study the latter, which are precisely the designs with constant replication and block size.We collect most known results, give new characterization results, and we enumerate, partly by computer, all small ones.matrices;singularities;geometry;combinatorics
New Classes of Partial Geometries and Their Associated LDPC Codes
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 . 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 figure
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