4,960 research outputs found
Good Codes From Generalised Algebraic Geometry Codes
Algebraic geometry codes or Goppa codes are defined with places of degree
one. In constructing generalised algebraic geometry codes places of higher
degree are used. In this paper we present 41 new codes over GF(16) which
improve on the best known codes of the same length and rate. The construction
method uses places of small degree with a technique originally published over
10 years ago for the construction of generalised algebraic geometry codes.Comment: 3 pages, to be presented at the IEEE Symposium on Information Theory
(ISIT 2010) in Austin, Texas, June 201
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
Cryptanalysis of public-key cryptosystems that use subcodes of algebraic geometry codes
We give a polynomial time attack on the McEliece public key cryptosystem
based on subcodes of algebraic geometry (AG) codes. The proposed attack reposes
on the distinguishability of such codes from random codes using the Schur
product. Wieschebrink treated the genus zero case a few years ago but his
approach cannot be extent straightforwardly to other genera. We address this
problem by introducing and using a new notion, which we call the t-closure of a
code
Two-Point Codes for the Generalized GK curve
We improve previously known lower bounds for the minimum distance of certain
two-point AG codes constructed using a Generalized Giulietti-Korchmaros curve
(GGK). Castellanos and Tizziotti recently described such bounds for two-point
codes coming from the Giulietti-Korchmaros curve (GK). Our results completely
cover and in many cases improve on their results, using different techniques,
while also supporting any GGK curve. Our method builds on the order bound for
AG codes: to enable this, we study certain Weierstrass semigroups. This allows
an efficient algorithm for computing our improved bounds. We find several new
improvements upon the MinT minimum distance tables.Comment: 13 page
Applications of finite geometry in coding theory and cryptography
We present in this article the basic properties of projective geometry, coding theory, and cryptography, and show how
finite geometry can contribute to coding theory and cryptography. In this way, we show links between three research areas, and in particular, show that finite geometry is not only interesting from a pure mathematical point of view, but also of interest for applications. We concentrate on introducing the basic concepts of these three research areas and give standard references for all these three research areas. We also mention particular results involving ideas from finite geometry, and particular results in cryptography involving ideas from coding theory
Information-theoretic Physical Layer Security for Satellite Channels
Shannon introduced the classic model of a cryptosystem in 1949, where Eve has
access to an identical copy of the cyphertext that Alice sends to Bob. Shannon
defined perfect secrecy to be the case when the mutual information between the
plaintext and the cyphertext is zero. Perfect secrecy is motivated by
error-free transmission and requires that Bob and Alice share a secret key.
Wyner in 1975 and later I.~Csisz\'ar and J.~K\"orner in 1978 modified the
Shannon model assuming that the channels are noisy and proved that secrecy can
be achieved without sharing a secret key. This model is called wiretap channel
model and secrecy capacity is known when Eve's channel is noisier than Bob's
channel.
In this paper we review the concept of wiretap coding from the satellite
channel viewpoint. We also review subsequently introduced stronger secrecy
levels which can be numerically quantified and are keyless unconditionally
secure under certain assumptions. We introduce the general construction of
wiretap coding and analyse its applicability for a typical satellite channel.
From our analysis we discuss the potential of keyless information theoretic
physical layer security for satellite channels based on wiretap coding. We also
identify system design implications for enabling simultaneous operation with
additional information theoretic security protocols
- …