3,325 research outputs found
A generalization of bounds for cyclic codes, including the HT and BS bounds
We use the algebraic structure of cyclic codes and some properties of the
discrete Fourier transform to give a reformulation of several classical bounds
for the distance of cyclic codes, by extending techniques of linear algebra. We
propose a bound, whose computational complexity is polynomial bounded, which is
a generalization of the Hartmann-Tzeng bound and the Betti-Sala bound. In the
majority of computed cases, our bound is the tightest among all known
polynomial-time bounds, including the Roos bound
Deterministic Construction of Binary, Bipolar and Ternary Compressed Sensing Matrices
In this paper we establish the connection between the Orthogonal Optical
Codes (OOC) and binary compressed sensing matrices. We also introduce
deterministic bipolar RIP fulfilling matrices of order
such that . The columns of these matrices are binary BCH code vectors where the
zeros are replaced by -1. Since the RIP is established by means of coherence,
the simple greedy algorithms such as Matching Pursuit are able to recover the
sparse solution from the noiseless samples. Due to the cyclic property of the
BCH codes, we show that the FFT algorithm can be employed in the reconstruction
methods to considerably reduce the computational complexity. In addition, we
combine the binary and bipolar matrices to form ternary sensing matrices
( elements) that satisfy the RIP condition.Comment: The paper is accepted for publication in IEEE Transaction on
Information Theor
Lemma for Linear Feedback Shift Registers and DFTs Applied to Affine Variety Codes
In this paper, we establish a lemma in algebraic coding theory that
frequently appears in the encoding and decoding of, e.g., Reed-Solomon codes,
algebraic geometry codes, and affine variety codes. Our lemma corresponds to
the non-systematic encoding of affine variety codes, and can be stated by
giving a canonical linear map as the composition of an extension through linear
feedback shift registers from a Grobner basis and a generalized inverse
discrete Fourier transform. We clarify that our lemma yields the error-value
estimation in the fast erasure-and-error decoding of a class of dual affine
variety codes. Moreover, we show that systematic encoding corresponds to a
special case of erasure-only decoding. The lemma enables us to reduce the
computational complexity of error-evaluation from O(n^3) using Gaussian
elimination to O(qn^2) with some mild conditions on n and q, where n is the
code length and q is the finite-field size.Comment: 37 pages, 1 column, 10 figures, 2 tables, resubmitted to IEEE
Transactions on Information Theory on Jan. 8, 201
Decoding of Projective Reed-Muller Codes by Dividing a Projective Space into Affine Spaces
A projective Reed-Muller (PRM) code, obtained by modifying a (classical)
Reed-Muller code with respect to a projective space, is a doubly extended
Reed-Solomon code when the dimension of the related projective space is equal
to 1. The minimum distance and dual code of a PRM code are known, and some
decoding examples have been represented for low-dimensional projective space.
In this study, we construct a decoding algorithm for all PRM codes by dividing
a projective space into a union of affine spaces. In addition, we determine the
computational complexity and the number of errors correctable of our algorithm.
Finally, we compare the codeword error rate of our algorithm with that of
minimum distance decoding.Comment: 17 pages, 4 figure
ELSI: A Unified Software Interface for Kohn-Sham Electronic Structure Solvers
Solving the electronic structure from a generalized or standard eigenproblem
is often the bottleneck in large scale calculations based on Kohn-Sham
density-functional theory. This problem must be addressed by essentially all
current electronic structure codes, based on similar matrix expressions, and by
high-performance computation. We here present a unified software interface,
ELSI, to access different strategies that address the Kohn-Sham eigenvalue
problem. Currently supported algorithms include the dense generalized
eigensolver library ELPA, the orbital minimization method implemented in
libOMM, and the pole expansion and selected inversion (PEXSI) approach with
lower computational complexity for semilocal density functionals. The ELSI
interface aims to simplify the implementation and optimal use of the different
strategies, by offering (a) a unified software framework designed for the
electronic structure solvers in Kohn-Sham density-functional theory; (b)
reasonable default parameters for a chosen solver; (c) automatic conversion
between input and internal working matrix formats, and in the future (d)
recommendation of the optimal solver depending on the specific problem.
Comparative benchmarks are shown for system sizes up to 11,520 atoms (172,800
basis functions) on distributed memory supercomputing architectures.Comment: 55 pages, 14 figures, 2 table
Design and Analysis of Nonbinary LDPC Codes for Arbitrary Discrete-Memoryless Channels
We present an analysis, under iterative decoding, of coset LDPC codes over
GF(q), designed for use over arbitrary discrete-memoryless channels
(particularly nonbinary and asymmetric channels). We use a random-coset
analysis to produce an effect that is similar to output-symmetry with binary
channels. We show that the random selection of the nonzero elements of the
GF(q) parity-check matrix induces a permutation-invariance property on the
densities of the decoder messages, which simplifies their analysis and
approximation. We generalize several properties, including symmetry and
stability from the analysis of binary LDPC codes. We show that under a Gaussian
approximation, the entire q-1 dimensional distribution of the vector messages
is described by a single scalar parameter (like the distributions of binary
LDPC messages). We apply this property to develop EXIT charts for our codes. We
use appropriately designed signal constellations to obtain substantial shaping
gains. Simulation results indicate that our codes outperform multilevel codes
at short block lengths. We also present simulation results for the AWGN
channel, including results within 0.56 dB of the unconstrained Shannon limit
(i.e. not restricted to any signal constellation) at a spectral efficiency of 6
bits/s/Hz.Comment: To appear, IEEE Transactions on Information Theory, (submitted
October 2004, revised and accepted for publication, November 2005). The
material in this paper was presented in part at the 41st Allerton Conference
on Communications, Control and Computing, October 2003 and at the 2005 IEEE
International Symposium on Information Theor
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