8,199 research outputs found
Unitary designs and codes
A unitary design is a collection of unitary matrices that approximate the
entire unitary group, much like a spherical design approximates the entire unit
sphere. In this paper, we use irreducible representations of the unitary group
to find a general lower bound on the size of a unitary t-design in U(d), for
any d and t. We also introduce the notion of a unitary code - a subset of U(d)
in which the trace inner product of any pair of matrices is restricted to only
a small number of distinct values - and give an upper bound for the size of a
code of degree s in U(d) for any d and s. These bounds can be strengthened when
the particular inner product values that occur in the code or design are known.
Finally, we describe some constructions of designs: we give an upper bound on
the size of the smallest weighted unitary t-design in U(d), and we catalogue
some t-designs that arise from finite groups.Comment: 25 pages, no figure
Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements
We introduce the problem of constructing weighted complex projective
2-designs from the union of a family of orthonormal bases. If the weight
remains constant across elements of the same basis, then such designs can be
interpreted as generalizations of complete sets of mutually unbiased bases,
being equivalent whenever the design is composed of d+1 bases in dimension d.
We show that, for the purpose of quantum state determination, these designs
specify an optimal collection of orthogonal measurements. Using highly
nonlinear functions on abelian groups, we construct explicit examples from d+2
orthonormal bases whenever d+1 is a prime power, covering dimensions d=6, 10,
and 12, for example, where no complete sets of mutually unbiased bases have
thus far been found.Comment: 28 pages, to appear in J. Math. Phy
Bounds for identifying codes in terms of degree parameters
An identifying code is a subset of vertices of a graph such that each vertex
is uniquely determined by its neighbourhood within the identifying code. If
\M(G) denotes the minimum size of an identifying code of a graph , it was
conjectured by F. Foucaud, R. Klasing, A. Kosowski and A. Raspaud that there
exists a constant such that if a connected graph with vertices and
maximum degree admits an identifying code, then \M(G)\leq
n-\tfrac{n}{d}+c. We use probabilistic tools to show that for any ,
\M(G)\leq n-\tfrac{n}{\Theta(d)} holds for a large class of graphs
containing, among others, all regular graphs and all graphs of bounded clique
number. This settles the conjecture (up to constants) for these classes of
graphs. In the general case, we prove \M(G)\leq n-\tfrac{n}{\Theta(d^{3})}.
In a second part, we prove that in any graph of minimum degree and
girth at least 5, \M(G)\leq(1+o_\delta(1))\tfrac{3\log\delta}{2\delta}n.
Using the former result, we give sharp estimates for the size of the minimum
identifying code of random -regular graphs, which is about
Filter Bank Fusion Frames
In this paper we characterize and construct novel oversampled filter banks
implementing fusion frames. A fusion frame is a sequence of orthogonal
projection operators whose sum can be inverted in a numerically stable way.
When properly designed, fusion frames can provide redundant encodings of
signals which are optimally robust against certain types of noise and erasures.
However, up to this point, few implementable constructions of such frames were
known; we show how to construct them using oversampled filter banks. In this
work, we first provide polyphase domain characterizations of filter bank fusion
frames. We then use these characterizations to construct filter bank fusion
frame versions of discrete wavelet and Gabor transforms, emphasizing those
specific finite impulse response filters whose frequency responses are
well-behaved.Comment: keywords: filter banks, frames, tight, fusion, erasures, polyphas
Coding for Errors and Erasures in Random Network Coding
The problem of error-control in random linear network coding is considered. A
``noncoherent'' or ``channel oblivious'' model is assumed where neither
transmitter nor receiver is assumed to have knowledge of the channel transfer
characteristic. Motivated by the property that linear network coding is
vector-space preserving, information transmission is modelled as the injection
into the network of a basis for a vector space and the collection by the
receiver of a basis for a vector space . A metric on the projective geometry
associated with the packet space is introduced, and it is shown that a minimum
distance decoder for this metric achieves correct decoding if the dimension of
the space is sufficiently large. If the dimension of each codeword
is restricted to a fixed integer, the code forms a subset of a finite-field
Grassmannian, or, equivalently, a subset of the vertices of the corresponding
Grassmann graph. Sphere-packing and sphere-covering bounds as well as a
generalization of the Singleton bound are provided for such codes. Finally, a
Reed-Solomon-like code construction, related to Gabidulin's construction of
maximum rank-distance codes, is described and a Sudan-style ``list-1'' minimum
distance decoding algorithm is provided.Comment: This revised paper contains some minor changes and clarification
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