2,177 research outputs found
Golay and other box codes
The (24,12;8) extended Golay Code can be generated as a 6 x 4 binary matrix from the (15,11;3) BCH-Hamming Code, represented as a 5 x 3 matrix, by adding a row and a column, both of odd or even parity. The odd-parity case provides the additional 12th dimension. Furthermore, any three columns and five rows of the 6 x 4 Golay form a BCH-Hamming (15,11;3) Code. Similarly a (80,58;8) code can be generated as a 10 x 8 binary matrix from the (63,57;3) BCH-Hamming Code represented as a 9 x 7 matrix by adding a row and a column both of odd and even parity. Furthermore, any seven columns along with the top nine rows is a BCH-Hamming (53,57;3) Code. A (80,40;16) 10 x 8 matrix binary code with weight structure identical to the extended (80,40;16) Quadratic Residue Code is generated from a (63,39;7) binary cyclic code represented as a 9 x 7 matrix, by adding a row and a column, both of odd or even parity
Algebraic techniques in designing quantum synchronizable codes
Quantum synchronizable codes are quantum error-correcting codes that can
correct the effects of quantum noise as well as block synchronization errors.
We improve the previously known general framework for designing quantum
synchronizable codes through more extensive use of the theory of finite fields.
This makes it possible to widen the range of tolerable magnitude of block
synchronization errors while giving mathematical insight into the algebraic
mechanism of synchronization recovery. Also given are families of quantum
synchronizable codes based on punctured Reed-Muller codes and their ambient
spaces.Comment: 9 pages, no figures. The framework presented in this article
supersedes the one given in arXiv:1206.0260 by the first autho
Numerical cubature using error-correcting codes
We present a construction for improving numerical cubature formulas with
equal weights and a convolution structure, in particular equal-weight product
formulas, using linear error-correcting codes. The construction is most
effective in low degree with extended BCH codes. Using it, we obtain several
sequences of explicit, positive, interior cubature formulas with good
asymptotics for each fixed degree as the dimension . Using a
special quadrature formula for the interval [arXiv:math.PR/0408360], we obtain
an equal-weight -cubature formula on the -cube with O(n^{\floor{t/2}})
points, which is within a constant of the Stroud lower bound. We also obtain
-cubature formulas on the -sphere, -ball, and Gaussian with
points when is odd. When is spherically symmetric and
, we obtain points. For each , we also obtain explicit,
positive, interior formulas for the -simplex with points; for
, we obtain O(n) points. These constructions asymptotically improve the
non-constructive Tchakaloff bound.
Some related results were recently found independently by Victoir, who also
noted that the basic construction more directly uses orthogonal arrays.Comment: Dedicated to Wlodzimierz and Krystyna Kuperberg on the occasion of
their 40th anniversary. This version has a major improvement for the n-cub
On the Peak-to-Mean Envelope Power Ratio of Phase-Shifted Binary Codes
The peak-to-mean envelope power ratio (PMEPR) of a code employed in
orthogonal frequency-division multiplexing (OFDM) systems can be reduced by
permuting its coordinates and by rotating each coordinate by a fixed phase
shift. Motivated by some previous designs of phase shifts using suboptimal
methods, the following question is considered in this paper. For a given binary
code, how much PMEPR reduction can be achieved when the phase shifts are taken
from a 2^h-ary phase-shift keying (2^h-PSK) constellation? A lower bound on the
achievable PMEPR is established, which is related to the covering radius of the
binary code. Generally speaking, the achievable region of the PMEPR shrinks as
the covering radius of the binary code decreases. The bound is then applied to
some well understood codes, including nonredundant BPSK signaling, BCH codes
and their duals, Reed-Muller codes, and convolutional codes. It is demonstrated
that most (presumably not optimal) phase-shift designs from the literature
attain or approach our bound.Comment: minor revisions, accepted for IEEE Trans. Commun
Stabilizer codes from modified symplectic form
Stabilizer codes form an important class of quantum error correcting codes
which have an elegant theory, efficient error detection, and many known
examples. Constructing stabilizer codes of length is equivalent to
constructing subspaces of which are
"isotropic" under the symplectic bilinear form defined by . As a
result, many, but not all, ideas from the theory of classical error correction
can be translated to quantum error correction. One of the main theoretical
contribution of this article is to study stabilizer codes starting with a
different symplectic form.
In this paper, we concentrate on cyclic codes. Modifying the symplectic form
allows us to generalize the previous known construction for linear cyclic
stabilizer codes, and in the process, circumvent some of the Galois theoretic
no-go results proved there. More importantly, this tweak in the symplectic form
allows us to make use of well known error correcting algorithms for cyclic
codes to give efficient quantum error correcting algorithms. Cyclicity of error
correcting codes is a "basis dependent" property. Our codes are no more
"cyclic" when they are derived using the standard symplectic forms (if we
ignore the error correcting properties like distance, all such symplectic forms
can be converted to each other via a basis transformation). Hence this change
of perspective is crucial from the point of view of designing efficient
decoding algorithm for these family of codes. In this context, recall that for
general codes, efficient decoding algorithms do not exist if some widely
believed complexity theoretic assumptions are true
Coding with Scrambling, Concatenation, and HARQ for the AWGN Wire-Tap Channel: A Security Gap Analysis
This study examines the use of nonsystematic channel codes to obtain secure
transmissions over the additive white Gaussian noise (AWGN) wire-tap channel.
Unlike the previous approaches, we propose to implement nonsystematic coded
transmission by scrambling the information bits, and characterize the bit error
rate of scrambled transmissions through theoretical arguments and numerical
simulations. We have focused on some examples of Bose-Chaudhuri-Hocquenghem
(BCH) and low-density parity-check (LDPC) codes to estimate the security gap,
which we have used as a measure of physical layer security, in addition to the
bit error rate. Based on a number of numerical examples, we found that such a
transmission technique can outperform alternative solutions. In fact, when an
eavesdropper (Eve) has a worse channel than the authorized user (Bob), the
security gap required to reach a given level of security is very small. The
amount of degradation of Eve's channel with respect to Bob's that is needed to
achieve sufficient security can be further reduced by implementing scrambling
and descrambling operations on blocks of frames, rather than on single frames.
While Eve's channel has a quality equal to or better than that of Bob's
channel, we have shown that the use of a hybrid automatic repeat-request (HARQ)
protocol with authentication still allows achieving a sufficient level of
security. Finally, the secrecy performance of some practical schemes has also
been measured in terms of the equivocation rate about the message at the
eavesdropper and compared with that of ideal codes.Comment: 29 pages, 10 figure
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