9,055 research outputs found
Group Divisible Codes and Their Application in the Construction of Optimal Constant-Composition Codes of Weight Three
The concept of group divisible codes, a generalization of group divisible
designs with constant block size, is introduced in this paper. This new class
of codes is shown to be useful in recursive constructions for constant-weight
and constant-composition codes. Large classes of group divisible codes are
constructed which enabled the determination of the sizes of optimal
constant-composition codes of weight three (and specified distance), leaving
only four cases undetermined. Previously, the sizes of constant-composition
codes of weight three were known only for those of sufficiently large length.Comment: 13 pages, 1 figure, 4 table
HFR Code: A Flexible Replication Scheme for Cloud Storage Systems
Fractional repetition (FR) codes are a family of repair-efficient storage
codes that provide exact and uncoded node repair at the minimum bandwidth
regenerating point. The advantageous repair properties are achieved by a
tailor-made two-layer encoding scheme which concatenates an outer
maximum-distance-separable (MDS) code and an inner repetition code. In this
paper, we generalize the application of FR codes and propose heterogeneous
fractional repetition (HFR) code, which is adaptable to the scenario where the
repetition degrees of coded packets are different. We provide explicit code
constructions by utilizing group divisible designs, which allow the design of
HFR codes over a large range of parameters. The constructed codes achieve the
system storage capacity under random access repair and have multiple repair
alternatives for node failures. Further, we take advantage of the systematic
feature of MDS codes and present a novel design framework of HFR codes, in
which storage nodes can be wisely partitioned into clusters such that data
reconstruction time can be reduced when contacting nodes in the same cluster.Comment: Accepted for publication in IET Communications, Jul. 201
Semifields, relative difference sets, and bent functions
Recently, the interest in semifields has increased due to the discovery of
several new families and progress in the classification problem. Commutative
semifields play an important role since they are equivalent to certain planar
functions (in the case of odd characteristic) and to modified planar functions
in even characteristic. Similarly, commutative semifields are equivalent to
relative difference sets. The goal of this survey is to describe the connection
between these concepts. Moreover, we shall discuss power mappings that are
planar and consider component functions of planar mappings, which may be also
viewed as projections of relative difference sets. It turns out that the
component functions in the even characteristic case are related to negabent
functions as well as to -valued bent functions.Comment: Survey paper for the RICAM workshop "Emerging applications of finite
fields", 09-13 December 2013, Linz, Austria. This article will appear in the
proceedings volume for this workshop, published as part of the "Radon Series
on Computational and Applied Mathematics" by DeGruyte
Divisible Design Graphs
AMS Subject Classification: 05B05, 05E30, 05C50.Strongly regular graph;Group divisible design;Deza graph;(v;k;)-Graph
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