527,099 research outputs found
Semidefinite programming bounds for Lee codes
For , let denote the maximum cardinality
of a code with minimum Lee distance at least ,
where denotes the cyclic group of order . We consider a
semidefinite programming bound based on triples of codewords, which bound can
be computed efficiently using symmetry reductions, resulting in several new
upper bounds on . The technique also yields an upper bound on the
independent set number of the -th strong product power of the circular graph
, which number is related to the Shannon capacity of . Here
is the graph with vertex set , in which two vertices
are adjacent if and only if their distance (mod ) is strictly less than .
The new bound does not seem to improve significantly over the bound obtained
from Lov\'asz theta-function, except for very small .Comment: 14 pages. arXiv admin note: text overlap with arXiv:1703.0517
Probabilistic lower bounds on maximal determinants of binary matrices
Let be the maximal determinant for -matrices, and be the ratio of
to the Hadamard upper bound. Using the probabilistic method,
we prove new lower bounds on and in terms of
, where is the order of a Hadamard matrix and is maximal
subject to . For example, if , and if . By a recent result of Livinskyi, as ,
so the second bound is close to for large . Previous
lower bounds tended to zero as with fixed, except in the
cases . For , our bounds are better for all
sufficiently large . If the Hadamard conjecture is true, then , so
the first bound above shows that is bounded below by a positive
constant .Comment: 17 pages, 2 tables, 24 references. Shorter version of
arXiv:1402.6817v4. Typos corrected in v2 and v3, new Lemma 7 in v4, updated
references in v5, added Remark 2.8 and a reference in v6, updated references
in v
On the number of coloured triangulations of -manifolds
We give superexponential lower and upper bounds on the number of coloured
-dimensional triangulations whose underlying space is an oriented manifold,
when the number of simplices goes to infinity and is fixed. In the
special case of dimension , the lower and upper bounds match up to
exponential factors, and we show that there are
coloured triangulations of -manifolds with tetrahedra. Our results also
imply that random coloured triangulations of -manifolds have a sublinear
number of vertices. Our upper bounds apply in particular to coloured
-spheres for which they seem to be the best known bounds in any dimension
, even though it is often conjectured that exponential bounds hold in
this case.
We also ask a related question on regular edge-coloured graphs having the
property that each -coloured component is planar, which is of independent
interest.Comment: 15 pages. New version, proof of the lower bound correcte
(Almost) tight bounds for randomized and quantum Local Search on hypercubes and grids
The Local Search problem, which finds a local minimum of a black-box function
on a given graph, is of both practical and theoretical importance to many areas
in computer science and natural sciences. In this paper, we show that for the
Boolean hypercube \B^n, the randomized query complexity of Local Search is
and the quantum query complexity is
. We also show that for the constant dimensional grid
, the randomized query complexity is for and the quantum query complexity is for . New
lower bounds for lower dimensional grids are also given. These improve the
previous results by Aaronson [STOC'04], and Santha and Szegedy [STOC'04].
Finally we show for a new upper bound of on the quantum query complexity, which implies that Local Search on
grids exhibits different properties at low dimensions.Comment: 18 pages, 1 figure. v2: introduction rewritten, references added. v3:
a line for grant added. v4: upper bound section rewritte
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