157 research outputs found
Semidefinite code bounds based on quadruple distances
Let be the maximum number of words of length , any two
having Hamming distance at least . We prove , which implies
that the quadruply shortened Golay code is optimal. Moreover, we show
, , , ,
, , , ,
, , , ,
, , and .
The method is based on the positive semidefiniteness of matrices derived from
quadruples of words. This can be put as constraint in a semidefinite program,
whose optimum value is an upper bound for . The order of the matrices
involved is huge. However, the semidefinite program is highly symmetric, by
which its feasible region can be restricted to the algebra of matrices
invariant under this symmetry. By block diagonalizing this algebra, the order
of the matrices will be reduced so as to make the program solvable with
semidefinite programming software in the above range of values of and .Comment: 15 page
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
The Independence Number of the Orthogonality Graph in Dimension
We determine the independence number of the orthogonality graph on
-dimensional hypercubes. This answers a question by Galliard from 2001
which is motivated by a problem in quantum information theory. Our method is a
modification of a rank argument due to Frankl who showed the analogous result
for -dimensional hypercubes, where is an odd prime.Comment: 3 pages, accepted by Combinatorica, fixed a minor typo spotted by
Peter Si
Semidefinite bounds for mixed binary/ternary codes
For nonnegative integers and , let denote the
maximum cardinality of a code of length , with binary
coordinates and ternary coordinates (in this order) and with minimum
distance at least . For a nonnegative integer , let
denote the collection of codes of cardinality at most . For , define . Then is upper bounded by the maximum
value of , where is a function
such that and if has minimum distance less than , and such that the matrix is positive semidefinite for each
. By exploiting symmetry, the semidefinite programming
problem for the case is reduced using representation theory. It yields
new upper bounds that are provided in tablesComment: 12 pages; some typos have been fixed. Accepted for publication in
Discrete Mathematic
Semidefinite bounds for nonbinary codes based on quadruples
For nonnegative integers , let denote the maximum
cardinality of a code of length over an alphabet with letters and
with minimum distance at least . We consider the following upper bound on
. For any , let \CC_k be the collection of codes of cardinality
at most . Then is at most the maximum value of
, where is a function \CC_4\to R_+ such that
and if has minimum distance less than , and
such that the \CC_2\times\CC_2 matrix (x(C\cup C'))_{C,C'\in\CC_2} is
positive semidefinite. By the symmetry of the problem, we can apply
representation theory to reduce the problem to a semidefinite programming
problem with order bounded by a polynomial in . It yields the new upper
bounds , , , and
Three-point bounds for energy minimization
Three-point semidefinite programming bounds are one of the most powerful
known tools for bounding the size of spherical codes. In this paper, we use
them to prove lower bounds for the potential energy of particles interacting
via a pair potential function. We show that our bounds are sharp for seven
points in RP^2. Specifically, we prove that the seven lines connecting opposite
vertices of a cube and of its dual octahedron are universally optimal. (In
other words, among all configurations of seven lines through the origin, this
one minimizes energy for all potential functions that are completely monotonic
functions of squared chordal distance.) This configuration is the only known
universal optimum that is not distance regular, and the last remaining
universal optimum in RP^2. We also give a new derivation of semidefinite
programming bounds and present several surprising conjectures about them.Comment: 30 page
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