65 research outputs found
Block diagonalization for algebra's associated with block codes
For a matrix *-algebra B, consider the matrix *-algebra A consisting of the
symmetric tensors in the n-fold tensor product of B. Examples of such algebras
in coding theory include the Bose-Mesner algebra and Terwilliger algebra of the
(non)binary Hamming cube, and algebras arising in SDP-hierarchies for coding
bounds using moment matrices. We give a computationally efficient block
diagonalization of A in terms of a given block diagonalization of B, and work
out some examples, including the Terwilliger algebra of the binary- and
nonbinary Hamming cube. As a tool we use some basic facts about representations
of the symmetric group.Comment: 16 page
On the Caratheodory rank of polymatroid bases
In this paper we prove that the Carath\'eodory rank of the set of bases of a
(poly)matroid is upper bounded by the cardinality of the ground set.Comment: 7 page
Polyhedra with the Integer Caratheodory Property
A polyhedron P has the Integer Caratheodory Property if the following holds.
For any positive integer k and any integer vector w in kP, there exist affinely
independent integer vectors x_1,...,x_t in P and positive integers n_1,...,n_t
such that n_1+...+n_t=k and w=n_1x_1+...+n_tx_t. In this paper we prove that if
P is a (poly)matroid base polytope or if P is defined by a TU matrix, then P
and projections of P satisfy the integer Caratheodory property.Comment: 12 page
Integer decomposition for polyhedra defined by nearly totally unimodular matrices
We call a matrix nearly totally unimodular if it can be obtained from a totally unimodular matrix by adding to each row of an integer multiple of some fixed row a^{\ssf T} of . For an integer vector and a nearly totally unimodular matrix , we denote by the integer hull of the set . We show that has the integer decomposition property and that we can find a decomposition of a given integer vector in polynomial time
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
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