77 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
A note on the stability number of an orthogonality graph
We consider the orthogonality graph Omega(n) with 2^n vertices corresponding
to the 0-1 n-vectors, two vertices adjacent if and only if the Hamming distance
between them is n/2. We show that the stability number of Omega(16) is
alpha(Omega(16))= 2304, thus proving a conjecture by Galliard. The main tool we
employ is a recent semidefinite programming relaxation for minimal distance
binary codes due to Schrijver.
As well, we give a general condition for Delsarte bound on the (co)cliques in
graphs of relations of association schemes to coincide with the ratio bound,
and use it to show that for Omega(n) the latter two bounds are equal to 2^n/n.Comment: 10 pages, LaTeX, 1 figure, companion Matlab code. Misc. misprints
fixed and references update
New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming
We give a new upper bound on the maximum size of a code of word length and minimum Hamming distance at least over the alphabet of letters. By block-diagonalizing the Terwilliger algebra of the nonbinary Hamming scheme, the bound can be calculated in time polynomial in using semidefinite programming. For this gives several improved upper bounds for concrete values of and . This work builds upon previous results of A. Schrijver [IEEE Trans. Inform. Theory 51 (2005), no. 8, 2859--2866] on the Terwilliger algebra of the binary Hamming schem
Entanglement of Free Fermions on Johnson Graphs
Free fermions on Johnson graphs are considered and the entanglement
entropy of sets of neighborhoods is computed. For a subsystem composed of a
single neighborhood, an analytical expression is provided by the decomposition
in irreducible submodules of the Terwilliger algebra of embedded in
two copies of . For a subsytem composed of multiple
neighborhoods, the construction of a block-tridiagonal operator which commutes
with the entanglement Hamiltonian is presented, its usefulness in computing the
entropy is stressed and the area law pre-factor is discussed.Comment: 24 page
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