17,725 research outputs found
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
A Geometric Theory for Hypergraph Matching
We develop a theory for the existence of perfect matchings in hypergraphs
under quite general conditions. Informally speaking, the obstructions to
perfect matchings are geometric, and are of two distinct types: 'space
barriers' from convex geometry, and 'divisibility barriers' from arithmetic
lattice-based constructions. To formulate precise results, we introduce the
setting of simplicial complexes with minimum degree sequences, which is a
generalisation of the usual minimum degree condition. We determine the
essentially best possible minimum degree sequence for finding an almost perfect
matching. Furthermore, our main result establishes the stability property:
under the same degree assumption, if there is no perfect matching then there
must be a space or divisibility barrier. This allows the use of the stability
method in proving exact results. Besides recovering previous results, we apply
our theory to the solution of two open problems on hypergraph packings: the
minimum degree threshold for packing tetrahedra in 3-graphs, and Fischer's
conjecture on a multipartite form of the Hajnal-Szemer\'edi Theorem. Here we
prove the exact result for tetrahedra and the asymptotic result for Fischer's
conjecture; since the exact result for the latter is technical we defer it to a
subsequent paper.Comment: Accepted for publication in Memoirs of the American Mathematical
Society. 101 pages. v2: minor changes including some additional diagrams and
passages of expository tex
On Some Properties of Quadratic APN Functions of a Special Form
In a recent paper, it is shown that functions of the form
, where and are linear, are a good source for
construction of new infinite families of APN functions. In the present work we
study necessary and sufficient conditions for such functions to be APN
A Multipartite Hajnal-Szemer\'edi Theorem
The celebrated Hajnal-Szemer\'edi theorem gives the precise minimum degree
threshold that forces a graph to contain a perfect K_k-packing. Fischer's
conjecture states that the analogous result holds for all multipartite graphs
except for those formed by a single construction. Recently, we deduced an
approximate version of this conjecture from new results on perfect matchings in
hypergraphs. In this paper, we apply a stability analysis to the extremal cases
of this argument, thus showing that the exact conjecture holds for any
sufficiently large graph.Comment: Final version, accepted to appear in JCTB. 43 pages, 2 figure
Proof of the 1-factorization and Hamilton decomposition conjectures III: approximate decompositions
In a sequence of four papers, we prove the following results (via a unified
approach) for all sufficiently large :
(i) [1-factorization conjecture] Suppose that is even and . Then every -regular graph on vertices has a
decomposition into perfect matchings. Equivalently, .
(ii) [Hamilton decomposition conjecture] Suppose that . Then every -regular graph on vertices has a decomposition
into Hamilton cycles and at most one perfect matching.
(iii) We prove an optimal result on the number of edge-disjoint Hamilton
cycles in a graph of given minimum degree.
According to Dirac, (i) was first raised in the 1950s. (ii) and (iii) answer
questions of Nash-Williams from 1970. The above bounds are best possible. In
the current paper, we show the following: suppose that is close to a
complete balanced bipartite graph or to the union of two cliques of equal size.
If we are given a suitable set of path systems which cover a set of
`exceptional' vertices and edges of , then we can extend these path systems
into an approximate decomposition of into Hamilton cycles (or perfect
matchings if appropriate).Comment: We originally split the proof into four papers, of which this was the
third paper. We have now combined this series into a single publication
[arXiv:1401.4159v2], which will appear in the Memoirs of the AMS. 29 pages, 2
figure
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