2,054 research outputs found
The Perfect Binary One-Error-Correcting Codes of Length 15: Part II--Properties
A complete classification of the perfect binary one-error-correcting codes of
length 15 as well as their extensions of length 16 was recently carried out in
[P. R. J. \"Osterg{\aa}rd and O. Pottonen, "The perfect binary
one-error-correcting codes of length 15: Part I--Classification," IEEE Trans.
Inform. Theory vol. 55, pp. 4657--4660, 2009]. In the current accompanying
work, the classified codes are studied in great detail, and their main
properties are tabulated. The results include the fact that 33 of the 80
Steiner triple systems of order 15 occur in such codes. Further understanding
is gained on full-rank codes via switching, as it turns out that all but two
full-rank codes can be obtained through a series of such transformations from
the Hamming code. Other topics studied include (non)systematic codes, embedded
one-error-correcting codes, and defining sets of codes. A classification of
certain mixed perfect codes is also obtained.Comment: v2: fixed two errors (extension of nonsystematic codes, table of
coordinates fixed by symmetries of codes), added and extended many other
result
Probabilistic existence of regular combinatorial structures
We show the existence of regular combinatorial objects which previously were
not known to exist. Specifically, for a wide range of the underlying
parameters, we show the existence of non-trivial orthogonal arrays, t-designs,
and t-wise permutations. In all cases, the sizes of the objects are optimal up
to polynomial overhead. The proof of existence is probabilistic. We show that a
randomly chosen structure has the required properties with positive yet tiny
probability. Our method allows also to give rather precise estimates on the
number of objects of a given size and this is applied to count the number of
orthogonal arrays, t-designs and regular hypergraphs. The main technical
ingredient is a special local central limit theorem for suitable lattice random
walks with finitely many steps.Comment: An extended abstract of this work [arXiv:1111.0492] appeared in STOC
2012. This version expands the literature discussio
Asymptotic enumeration of correlation-immune boolean functions
A boolean function of boolean variables is {correlation-immune} of order
if the function value is uncorrelated with the values of any of the
arguments. Such functions are of considerable interest due to their
cryptographic properties, and are also related to the orthogonal arrays of
statistics and the balanced hypercube colourings of combinatorics. The {weight}
of a boolean function is the number of argument values that produce a function
value of 1. If this is exactly half the argument values, that is,
values, a correlation-immune function is called {resilient}.
An asymptotic estimate of the number of -variable
correlation-immune boolean functions of order was obtained in 1992 by
Denisov for constant . Denisov repudiated that estimate in 2000, but we will
show that the repudiation was a mistake.
The main contribution of this paper is an asymptotic estimate of
which holds if increases with within generous limits and specialises to
functions with a given weight, including the resilient functions. In the case
of , our estimates are valid for all weights.Comment: 18 page
Efficient Decoupling Schemes Based on Hamilton Cycles
Decoupling the interactions in a spin network governed by a pair-interaction
Hamiltonian is a well-studied problem. Combinatorial schemes for decoupling and
for manipulating the couplings of Hamiltonians have been developed which use
selective pulses. In this paper we consider an additional requirement on these
pulse sequences: as few {\em different} control operations as possible should
be used. This requirement is motivated by the fact that optimizing each
individual selective pulse will be expensive, i. e., it is desirable to use as
few different selective pulses as possible. For an arbitrary -dimensional
system we show that the ability to implement only two control operations is
sufficient to turn off the time evolution. In case of a bipartite system with
local control we show that four different control operations are sufficient.
Turning to networks consisting of several -dimensional nodes which are
governed by a pair-interaction Hamiltonian, we show that decoupling can be
achieved if one is able to control a number of different control operations
which is logarithmic in the number of nodes.Comment: 4 pages, 1 figure, uses revtex
The Trapping Redundancy of Linear Block Codes
We generalize the notion of the stopping redundancy in order to study the
smallest size of a trapping set in Tanner graphs of linear block codes. In this
context, we introduce the notion of the trapping redundancy of a code, which
quantifies the relationship between the number of redundant rows in any
parity-check matrix of a given code and the size of its smallest trapping set.
Trapping sets with certain parameter sizes are known to cause error-floors in
the performance curves of iterative belief propagation decoders, and it is
therefore important to identify decoding matrices that avoid such sets. Bounds
on the trapping redundancy are obtained using probabilistic and constructive
methods, and the analysis covers both general and elementary trapping sets.
Numerical values for these bounds are computed for the [2640,1320] Margulis
code and the class of projective geometry codes, and compared with some new
code-specific trapping set size estimates.Comment: 12 pages, 4 tables, 1 figure, accepted for publication in IEEE
Transactions on Information Theor
A trigonometric approach to quaternary code designs with application to one-eighth and one-sixteenth fractions
The study of good nonregular fractional factorial designs has received
significant attention over the last two decades. Recent research indicates that
designs constructed from quaternary codes (QC) are very promising in this
regard. The present paper shows how a trigonometric approach can facilitate a
systematic understanding of such QC designs and lead to new theoretical results
covering hitherto unexplored situations. We focus attention on one-eighth and
one-sixteenth fractions of two-level factorials and show that optimal QC
designs often have larger generalized resolution and projectivity than
comparable regular designs. Moreover, some of these designs are found to have
maximum projectivity among all designs.Comment: Published in at http://dx.doi.org/10.1214/10-AOS815 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Quarter-fraction factorial designs constructed via quaternary codes
The research of developing a general methodology for the construction of good
nonregular designs has been very active in the last decade. Recent research by
Xu and Wong [Statist. Sinica 17 (2007) 1191--1213] suggested a new class of
nonregular designs constructed from quaternary codes. This paper explores the
properties and uses of quaternary codes toward the construction of
quarter-fraction nonregular designs. Some theoretical results are obtained
regarding the aliasing structure of such designs. Optimal designs are
constructed under the maximum resolution, minimum aberration and maximum
projectivity criteria. These designs often have larger generalized resolution
and larger projectivity than regular designs of the same size. It is further
shown that some of these designs have generalized minimum aberration and
maximum projectivity among all possible designs.Comment: Published in at http://dx.doi.org/10.1214/08-AOS656 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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