7,253 research outputs found
Near-complete external difference families
We introduce and explore near-complete external difference families, a partitioning of the nonidentity elements of a group so that each nonidentity element is expressible as a difference of elements from distinct subsets a fixed number of times. We show that the existence of such an object implies the existence of a near-resolvable design. We provide examples and general constructions of these objects, some of which lead to new parameter families of near-resolvable designs on a non-prime-power number of points. Our constructions employ cyclotomy, partial difference sets, and Galois rings.PostprintPeer reviewe
Frame difference families and resolvable balanced incomplete block designs
Frame difference families, which can be obtained via a careful use of
cyclotomic conditions attached to strong difference families, play an important
role in direct constructions for resolvable balanced incomplete block designs.
We establish asymptotic existences for several classes of frame difference
families. As corollaries new infinite families of 1-rotational
-RBIBDs over are
derived, and the existence of -RBIBDs is discussed. We construct
-RBIBDs for , whose
existence were previously in doubt. As applications, we establish asymptotic
existences for an infinite family of optimal constant composition codes and an
infinite family of strictly optimal frequency hopping sequences.Comment: arXiv admin note: text overlap with arXiv:1702.0750
Efficient Two-Stage Group Testing Algorithms for Genetic Screening
Efficient two-stage group testing algorithms that are particularly suited for
rapid and less-expensive DNA library screening and other large scale biological
group testing efforts are investigated in this paper. The main focus is on
novel combinatorial constructions in order to minimize the number of individual
tests at the second stage of a two-stage disjunctive testing procedure.
Building on recent work by Levenshtein (2003) and Tonchev (2008), several new
infinite classes of such combinatorial designs are presented.Comment: 14 pages; to appear in "Algorithmica". Part of this work has been
presented at the ICALP 2011 Group Testing Workshop; arXiv:1106.368
Fractional repetition codes with flexible repair from combinatorial designs
Fractional repetition (FR) codes are a class of regenerating codes for
distributed storage systems with an exact (table-based) repair process that is
also uncoded, i.e., upon failure, a node is regenerated by simply downloading
packets from the surviving nodes. In our work, we present constructions of FR
codes based on Steiner systems and resolvable combinatorial designs such as
affine geometries, Hadamard designs and mutually orthogonal Latin squares. The
failure resilience of our codes can be varied in a simple manner. We construct
codes with normalized repair bandwidth () strictly larger than one;
these cannot be obtained trivially from codes with . Furthermore, we
present the Kronecker product technique for generating new codes from existing
ones and elaborate on their properties. FR codes with locality are those where
the repair degree is smaller than the number of nodes contacted for
reconstructing the stored file. For these codes we establish a tradeoff between
the local repair property and failure resilience and construct codes that meet
this tradeoff. Much of prior work only provided lower bounds on the FR code
rate. In our work, for most of our constructions we determine the code rate for
certain parameter ranges.Comment: 27 pages in IEEE two-column format. IEEE Transactions on Information
Theory (to appear
New Combinatorial Construction Techniques for Low-Density Parity-Check Codes and Systematic Repeat-Accumulate Codes
This paper presents several new construction techniques for low-density
parity-check (LDPC) and systematic repeat-accumulate (RA) codes. Based on
specific classes of combinatorial designs, the improved code design focuses on
high-rate structured codes with constant column weights 3 and higher. The
proposed codes are efficiently encodable and exhibit good structural
properties. Experimental results on decoding performance with the sum-product
algorithm show that the novel codes offer substantial practical application
potential, for instance, in high-speed applications in magnetic recording and
optical communications channels.Comment: 10 pages; to appear in "IEEE Transactions on Communications
Adjoining a universal inner inverse to a ring element
Let be an associative unital algebra over a field let be an
element of and let We obtain normal
forms for elements of and for elements of -modules arising by
extension of scalars from -modules. The details depend on where in the chain
the unit of
first appears.
This investigation is motivated by a hoped-for application to the study of
the possible forms of the monoid of isomorphism classes of finitely generated
projective modules over a von Neumann regular ring; but that goal remains
distant.
We end with a normal form result for the algebra obtained by tying together a
-algebra given with a nonzero element satisfying and
a -algebra given with a nonzero satisfying via the
pair of relations Comment: 28 pages. Results on mutual inner inverses added at end of earlier
version, and much clarification of wording etc.. After publication, any
updates, errata, related references etc. found will be recorded at
http://math.berkeley.edu/~gbergman/paper
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