460 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
Constructions of Pure Asymmetric Quantum Alternant Codes Based on Subclasses of Alternant Codes
In this paper, we construct asymmetric quantum error-correcting codes(AQCs)
based on subclasses of Alternant codes. Firstly, We propose a new subclass of
Alternant codes which can attain the classical Gilbert-Varshamov bound to
construct AQCs. It is shown that when , -parts of the AQCs can attain
the classical Gilbert-Varshamov bound. Then we construct AQCs based on a famous
subclass of Alternant codes called Goppa codes. As an illustrative example, we
get three AQCs from the well
known binary Goppa code. At last, we get asymptotically good
binary expansions of asymmetric quantum GRS codes, which are quantum
generalizations of Retter's classical results. All the AQCs constructed in this
paper are pure
New Identities Relating Wild Goppa Codes
For a given support and a polynomial with no roots in , we prove equality
between the -ary Goppa codes where
denotes the norm of , that is In
particular, for , that is, for a quadratic extension, we get
. If has roots in
, then we do not necessarily have equality and we prove that
the difference of the dimensions of the two codes is bounded above by the
number of distinct roots of in . These identities provide
numerous code equivalences and improved designed parameters for some families
of classical Goppa codes.Comment: 14 page
Coding Solutions for the Secure Biometric Storage Problem
The paper studies the problem of securely storing biometric passwords, such
as fingerprints and irises. With the help of coding theory Juels and Wattenberg
derived in 1999 a scheme where similar input strings will be accepted as the
same biometric. In the same time nothing could be learned from the stored data.
They called their scheme a "fuzzy commitment scheme". In this paper we will
revisit the solution of Juels and Wattenberg and we will provide answers to two
important questions: What type of error-correcting codes should be used and
what happens if biometric templates are not uniformly distributed, i.e. the
biometric data come with redundancy. Answering the first question will lead us
to the search for low-rate large-minimum distance error-correcting codes which
come with efficient decoding algorithms up to the designed distance. In order
to answer the second question we relate the rate required with a quantity
connected to the "entropy" of the string, trying to estimate a sort of
"capacity", if we want to see a flavor of the converse of Shannon's noisy
coding theorem. Finally we deal with side-problems arising in a practical
implementation and we propose a possible solution to the main one that seems to
have so far prevented real life applications of the fuzzy scheme, as far as we
know.Comment: the final version appeared in Proceedings Information Theory Workshop
(ITW) 2010, IEEE copyrigh
On spectra of BCH codes
Derives an estimate for the error term in the binomial approximation of spectra of BCH codes. This estimate asymptotically improves on the bounds by Sidelnikov (1971), Kasami et al. (1985), and Sole (1990)
Quantum Fourier sampling, Code Equivalence, and the quantum security of the McEliece and Sidelnikov cryptosystems
The Code Equivalence problem is that of determining whether two given linear
codes are equivalent to each other up to a permutation of the coordinates. This
problem has a direct reduction to a nonabelian hidden subgroup problem (HSP),
suggesting a possible quantum algorithm analogous to Shor's algorithms for
factoring or discrete log. However, we recently showed that in many cases of
interest---including Goppa codes---solving this case of the HSP requires rich,
entangled measurements. Thus, solving these cases of Code Equivalence via
Fourier sampling appears to be out of reach of current families of quantum
algorithms.
Code equivalence is directly related to the security of McEliece-type
cryptosystems in the case where the private code is known to the adversary.
However, for many codes the support splitting algorithm of Sendrier provides a
classical attack in this case. We revisit the claims of our previous article in
the light of these classical attacks, and discuss the particular case of the
Sidelnikov cryptosystem, which is based on Reed-Muller codes
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