1,096 research outputs found
Decoder for Nonbinary CWS Quantum Codes
We present a decoder for nonbinary CWS quantum codes using the structure of
union codes. The decoder runs in two steps: first we use a union of stabilizer
codes to detect a sequence of errors, and second we build a new code, called
union code, that allows to correct the errors
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
On the Parameters of Convolutional Codes with Cyclic Structure
In this paper convolutional codes with cyclic structure will be investigated.
These codes can be understood as left principal ideals in a suitable
skew-polynomial ring. It has been shown in [3] that only certain combinations
of the parameters (field size, length, dimension, and Forney indices) can occur
for cyclic codes. We will investigate whether all these combinations can indeed
be realized by a suitable cyclic code and, if so, how to construct such a code.
A complete characterization and construction will be given for minimal cyclic
codes. It is derived from a detailed investigation of the units in the
skew-polynomial ring
Difference Sets and Positive Exponential Sums I. General Properties
We describe general connections between intersective properties of sets in Abelian groups and positive exponential sums. In particular, given a set A the maximal size of a set whose difference set avoids A will be related to positive exponential sums using frequencies from A. Ā© 2013 Springer Science+Business Media New York
Difference sets and positive exponential sums I. General properties
We describe general connections between intersective properties of sets in
Abelian groups and positive exponential sums. In particular, given a set
the maximal size of a set whose difference set avoids will be related to
positive exponential sums using frequencies from .Comment: 21 pages, a few remarks added, and one theorem slightly extende
Trellis Computations
For a certain class of functions, the distribution of the function values can
be calculated in the trellis or a sub-trellis. The forward/backward recursion
known from the BCJR algorithm is generalized to compute the moments of these
distributions. In analogy to the symbol probabilities, by introducing a
constraint at a certain depth in the trellis we obtain symbol moments. These
moments are required for an efficient implementation of the discriminated
belief propagation algorithm in [2], and can furthermore be utilized to compute
conditional entropies in the trellis.
The moment computation algorithm has the same asymptotic complexity as the
BCJR algorithm. It is applicable to any commutative semi-ring, thus actually
providing a generalization of the Viterbi algorithm.Comment: 9 pages, 4 figure
The combinatorics of LCD codes: Linear Programming bound and orthogonal matrices
Linear Complementary Dual codes (LCD) are binary linear codes that meet their
dual trivially. We construct LCD codes using orthogonal matrices, self-dual
codes, combinatorial designs and Gray map from codes over the family of rings
. We give a linear programming bound on the largest size of an LCD code of
given length and minimum distance. We make a table of lower bounds for this
combinatorial function for modest values of the parameters.Comment: submitted to Linear Algebra and Applications on June, 1, 201
On doubly-cyclic convolutional codes
Cyclicity of a convolutional code (CC) is relying on a nontrivial
automorphism of the algebra F[x]/(x^n-1), where F is a finite field. If this
automorphism itself has certain specific cyclicity properties one is lead to
the class of doubly-cyclic CC's. Within this large class Reed-Solomon and BCH
convolutional codes can be defined. After constructing doubly-cyclic CC's,
basic properties are derived on the basis of which distance properties of
Reed-Solomon convolutional codes are investigated.This shows that some of them
are optimal or near optimal with respect to distance and performance
Discrete phase-space structure of -qubit mutually unbiased bases
We work out the phase-space structure for a system of qubits. We replace
the field of real numbers that label the axes of the continuous phase space by
the finite field \Gal{2^n} and investigate the geometrical structures
compatible with the notion of unbiasedness. These consist of bundles of
discrete curves intersecting only at the origin and satisfying certain
additional properties. We provide a simple classification of such curves and
study in detail the four- and eight-dimensional cases, analyzing also the
effect of local transformations. In this way, we provide a comprehensive
phase-space approach to the construction of mutually unbiased bases for
qubits.Comment: Title changed. Improved version. Accepted for publication in Annals
of Physic
Entropy operates in non-linear semifields
We set out to demonstrate that the R\'enyi entropies with parameter
are better thought of as operating in a type of non-linear semiring called a
positive semifield. We show how the R\'enyi's postulates lead to Pap's
g-calculus where the functions carrying out the domain transformation are
Renyi's information function and its inverse. In its turn, Pap's g-calculus
under R\'enyi's information function transforms the set of positive reals into
a family of semirings where "standard" product has been transformed into sum
and "standard" sum into a power-deformed sum. Consequently, the transformed
product has an inverse whence the structure is actually that of a positive
semifield. Instances of this construction lead into idempotent analysis and
tropical algebra as well as to less exotic structures. Furthermore, shifting
the definition of the parameter shows in full the intimate relation of
the R\'enyi entropies to the weighted generalized power means. We conjecture
that this is one of the reasons why tropical algebra procedures, like the
Viterbi algorithm of dynamic programming, morphological processing, or neural
networks are so successful in computational intelligence applications. But
also, why there seem to exist so many procedures to deal with "information" at
large.Comment: Version 2.0. Excised the case for shifting the R\'enyi entropy (see
sibling paper). 20 pages, 3 figures, 2 table
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