10,905 research outputs found
Constructions of Generalized Sidon Sets
We give explicit constructions of sets S with the property that for each
integer k, there are at most g solutions to k=s_1+s_2, s_i\in S; such sets are
called Sidon sets if g=2 and generalized Sidon sets if g\ge 3. We extend to
generalized Sidon sets the Sidon-set constructions of Singer, Bose, and Ruzsa.
We also further optimize Koulantzakis' idea of interleaving several copies of a
Sidon set, extending the improvements of Cilleruelo & Ruzsa & Trujillo, Jia,
and Habsieger & Plagne. The resulting constructions yield the largest known
generalized Sidon sets in virtually all cases.Comment: 15 pages, 1 figure (revision fixes typos, adds a few details, and
adjusts notation
Load-Balanced Fractional Repetition Codes
We introduce load-balanced fractional repetition (LBFR) codes, which are a
strengthening of fractional repetition (FR) codes. LBFR codes have the
additional property that multiple node failures can be sequentially repaired by
downloading no more than one block from any other node. This allows for better
use of the network, and can additionally reduce the number of disk reads
necessary to repair multiple nodes. We characterize LBFR codes in terms of
their adjacency graphs, and use this characterization to present explicit
constructions LBFR codes with storage capacity comparable existing FR codes.
Surprisingly, in some parameter regimes, our constructions of LBFR codes match
the parameters of the best constructions of FR codes
Covering of Subspaces by Subspaces
Lower and upper bounds on the size of a covering of subspaces in the
Grassmann graph \cG_q(n,r) by subspaces from the Grassmann graph
\cG_q(n,k), , are discussed. The problem is of interest from four
points of view: coding theory, combinatorial designs, -analogs, and
projective geometry. In particular we examine coverings based on lifted maximum
rank distance codes, combined with spreads and a recursive construction. New
constructions are given for with or . We discuss the density
for some of these coverings. Tables for the best known coverings, for and
, are presented. We present some questions concerning
possible constructions of new coverings of smaller size.Comment: arXiv admin note: text overlap with arXiv:0805.352
High-rate self-synchronizing codes
Self-synchronization under the presence of additive noise can be achieved by
allocating a certain number of bits of each codeword as markers for
synchronization. Difference systems of sets are combinatorial designs which
specify the positions of synchronization markers in codewords in such a way
that the resulting error-tolerant self-synchronizing codes may be realized as
cosets of linear codes. Ideally, difference systems of sets should sacrifice as
few bits as possible for a given code length, alphabet size, and
error-tolerance capability. However, it seems difficult to attain optimality
with respect to known bounds when the noise level is relatively low. In fact,
the majority of known optimal difference systems of sets are for exceptionally
noisy channels, requiring a substantial amount of bits for synchronization. To
address this problem, we present constructions for difference systems of sets
that allow for higher information rates while sacrificing optimality to only a
small extent. Our constructions utilize optimal difference systems of sets as
ingredients and, when applied carefully, generate asymptotically optimal ones
with higher information rates. We also give direct constructions for optimal
difference systems of sets with high information rates and error-tolerance that
generate binary and ternary self-synchronizing codes.Comment: 9 pages, no figure, 2 tables. Final accepted version for publication
in the IEEE Transactions on Information Theory. Material presented in part at
the International Symposium on Information Theory and its Applications,
Honolulu, HI USA, October 201
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