2 research outputs found
Performance and analysis of Quadratic Residue Codes of lengths less than 100
In this paper, the performance of quadratic residue (QR) codes of lengths
within 100 is given and analyzed when the hard decoding, soft decoding, and
linear programming decoding algorithms are utilized. We develop a simple method
to estimate the soft decoding performance, which avoids extensive simulations.
Also, a simulation-based algorithm is proposed to obtain the maximum likelihood
decoding performance of QR codes of lengths within 100. Moreover, four
important theorems are proposed to predict the performance of the hard decoding
and the maximum-likelihood decoding in which they can explore some internal
properties of QR codes. It is shown that such four theorems can be applied to
the QR codes with lengths less than 100 for predicting the decoding
performance. In contrast, they can be straightforwardly generalized to longer
QR codes. The result is never seen in the literature, to our knowledge.
Simulation results show that the estimated hard decoding performance is very
accurate in the whole signal-to-noise ratio (SNR) regimes, whereas the derived
upper bounds of the maximum likelihood decoding are only tight for moderate to
high SNR regions. For each of the considered QR codes, the soft decoding is
approximately 1.5 dB better than the hard decoding. By using powerful redundant
parity-check cuts, the linear programming-based decoding algorithm, i.e., the
ACG-ALP decoding algorithm performs very well for any QR code. Sometimes, it is
even superior to the Chase-based soft decoding algorithm significantly, and
hence is only a few tenths of dB away from the maximum likelihood decoding.Comment: submitted to IEEE Transactions on Information Theor
A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices
In quantum coding theory, stabilizer codes are probably the most important
class of quantum codes. They are regarded as the quantum analogue of the
classical linear codes and the properties of stabilizer codes have been
carefully studied in the literature. In this paper, a new but simple
construction of stabilizer codes is proposed based on syndrome assignment by
classical parity-check matrices. This method reduces the construction of
quantum stabilizer codes to the construction of classical parity-check matrices
that satisfy a specific commutative condition. The quantum stabilizer codes
from this construction have a larger set of correctable error operators than
expected. Its (asymptotic) coding efficiency is comparable to that of CSS
codes. A class of quantum Reed-Muller codes is constructed, which have a larger
set of correctable error operators than that of the quantum Reed-Muller codes
developed previously in the literature. Quantum stabilizer codes inspired by
classical quadratic residue codes are also constructed and some of which are
optimal in terms of their coding parameters.Comment: 34 pages, 3 figures, 5 tables, index terms add, abstract and
conclusion slightly modifie