931 research outputs found
Algebraic Codes For Error Correction In Digital Communication Systems
Access to the full-text thesis is no longer available at the author's request, due to 3rd party copyright restrictions. Access removed on 29.11.2016 by CS (TIS).Metadata merged with duplicate record (http://hdl.handle.net/10026.1/899) on 20.12.2016 by CS (TIS).C. Shannon presented theoretical conditions under which communication was possible
error-free in the presence of noise. Subsequently the notion of using error
correcting codes to mitigate the effects of noise in digital transmission was introduced
by R. Hamming. Algebraic codes, codes described using powerful tools from
algebra took to the fore early on in the search for good error correcting codes. Many
classes of algebraic codes now exist and are known to have the best properties of
any known classes of codes. An error correcting code can be described by three of its
most important properties length, dimension and minimum distance. Given codes
with the same length and dimension, one with the largest minimum distance will
provide better error correction. As a result the research focuses on finding improved
codes with better minimum distances than any known codes.
Algebraic geometry codes are obtained from curves. They are a culmination of years
of research into algebraic codes and generalise most known algebraic codes. Additionally
they have exceptional distance properties as their lengths become arbitrarily
large. Algebraic geometry codes are studied in great detail with special attention
given to their construction and decoding. The practical performance of these codes
is evaluated and compared with previously known codes in different communication
channels. Furthermore many new codes that have better minimum distance
to the best known codes with the same length and dimension are presented from
a generalised construction of algebraic geometry codes. Goppa codes are also an
important class of algebraic codes. A construction of binary extended Goppa codes
is generalised to codes with nonbinary alphabets and as a result many new codes
are found. This construction is shown as an efficient way to extend another well
known class of algebraic codes, BCH codes. A generic method of shortening codes
whilst increasing the minimum distance is generalised. An analysis of this method
reveals a close relationship with methods of extending codes. Some new codes from
Goppa codes are found by exploiting this relationship. Finally an extension method
for BCH codes is presented and this method is shown be as good as a well known
method of extension in certain cases
On a Class of Optimal Nonbinary Linear Unequal-Error-Protection Codes for Two Sets of Messages
Several authors have addressed the problem of designing good linear unequal error protection (LUEP) codes. However, very little is known about good nonbinary LUEP codes. We present a class of optimal nonbinary LUEP codes for two different sets of messages. By combining t-error-correcting ReedSolomon (RS) codes and shortened nonbinary Hamming codes, we obtain nonbinary LUEP codes that protect one set of messages against any t or fewer symbol errors and the remaining set of messages against any single symbol error. For t ≥ 2, we show that these codes are optimal in the sense of achieving the Hamming lower bound on the number of redundant symbols of a nonbinary LUEP code with the same parameters
Scheme for constructing graphs associated with stabilizer quantum codes
We propose a systematic scheme for the construction of graphs associated with
binary stabilizer codes. The scheme is characterized by three main steps:
first, the stabilizer code is realized as a codeword-stabilized (CWS) quantum
code; second, the canonical form of the CWS code is uncovered; third, the input
vertices are attached to the graphs. To check the effectiveness of the scheme,
we discuss several graphical constructions of various useful stabilizer codes
characterized by single and multi-qubit encoding operators. In particular, the
error-correcting capabilities of such quantum codes are verified in
graph-theoretic terms as originally advocated by Schlingemann and Werner.
Finally, possible generalizations of our scheme for the graphical construction
of both (stabilizer and nonadditive) nonbinary and continuous-variable quantum
codes are briefly addressed.Comment: 42 pages, 12 figure
Graphical Nonbinary Quantum Error-Correcting Codes
In this paper, based on the nonbinary graph state, we present a systematic
way of constructing good non-binary quantum codes, both additive and
nonadditive, for systems with integer dimensions. With the help of computer
search, which results in many interesting codes including some nonadditive
codes meeting the Singleton bounds, we are able to construct explicitly four
families of optimal codes, namely, , ,
and for any odd dimension and a family of nonadditive code
for arbitrary . In the case of composite numbers as
dimensions, we also construct a family of stabilizer codes for odd , whose coding subspace is {\em not} of a dimension
that is a power of the dimension of the physical subsystem.Comment: 12 pages, 5 figures (pdf
Correcting a Fraction of Errors in Nonbinary Expander Codes with Linear Programming
A linear-programming decoder for \emph{nonbinary} expander codes is
presented. It is shown that the proposed decoder has the maximum-likelihood
certificate properties. It is also shown that this decoder corrects any pattern
of errors of a relative weight up to approximately 1/4 \delta_A \delta_B (where
\delta_A and \delta_B are the relative minimum distances of the constituent
codes).Comment: Part of this work was presented at the IEEE International Symposium
on Information Theory 2009, Seoul, Kore
Long Nonbinary Codes Exceeding the Gilbert - Varshamov Bound for any Fixed Distance
Let A(q,n,d) denote the maximum size of a q-ary code of length n and distance
d. We study the minimum asymptotic redundancy \rho(q,n,d)=n-log_q A(q,n,d) as n
grows while q and d are fixed. For any d and q<=d-1, long algebraic codes are
designed that improve on the BCH codes and have the lowest asymptotic
redundancy \rho(q,n,d) <= ((d-3)+1/(d-2)) log_q n known to date. Prior to this
work, codes of fixed distance that asymptotically surpass BCH codes and the
Gilbert-Varshamov bound were designed only for distances 4,5 and 6.Comment: Submitted to IEEE Trans. on Info. Theor
Algebraic geometric construction of a quantum stabilizer code
The stabilizer code is the most general algebraic construction of quantum
error-correcting codes proposed so far. A stabilizer code can be constructed
from a self-orthogonal subspace of a symplectic space over a finite field. We
propose a construction method of such a self-orthogonal space using an
algebraic curve. By using the proposed method we construct an asymptotically
good sequence of binary stabilizer codes. As a byproduct we improve the
Ashikhmin-Litsyn-Tsfasman bound of quantum codes. The main results in this
paper can be understood without knowledge of quantum mechanics.Comment: LaTeX2e, 12 pages, 1 color figure. A decoding method was added and
several typographical errors were corrected in version 2. The description of
the decoding problem was completely wrong in version 1. In version 1 and 2,
there was a critical miscalculation in the estimation of parameters of codes,
and the constructed sequence of codes turned out to be worse than existing
ones. The asymptotically best sequence of quantum codes was added in version
3. Section 3.2 appeared in IEEE Transactions on Information Theory, vol. 48,
no. 7, pp. 2122-2124, July 200
Neural networks, error-correcting codes, and polynomials over the binary n-cube
Several ways of relating the concept of error-correcting codes to the concept of neural networks are presented. Performing maximum-likelihood decoding in a linear block error-correcting code is shown to be equivalent to finding a global maximum of the energy function of a certain neural network. Given a linear block code, a neural network can be constructed in such a way that every codeword corresponds to a local maximum. The connection between maximization of polynomials over the n-cube and error-correcting codes is also investigated; the results suggest that decoding techniques can be a useful tool for solving such maximization problems. The results are generalized to both nonbinary and nonlinear codes
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