166,918 research outputs found
Algebraic geometry codes with complementary duals exceed the asymptotic Gilbert-Varshamov bound
It was shown by Massey that linear complementary dual (LCD for short) codes
are asymptotically good. In 2004, Sendrier proved that LCD codes meet the
asymptotic Gilbert-Varshamov (GV for short) bound. Until now, the GV bound
still remains to be the best asymptotical lower bound for LCD codes. In this
paper, we show that an algebraic geometry code over a finite field of even
characteristic is equivalent to an LCD code and consequently there exists a
family of LCD codes that are equivalent to algebraic geometry codes and exceed
the asymptotical GV bound
KCRC-LCD: Discriminative Kernel Collaborative Representation with Locality Constrained Dictionary for Visual Categorization
We consider the image classification problem via kernel collaborative
representation classification with locality constrained dictionary (KCRC-LCD).
Specifically, we propose a kernel collaborative representation classification
(KCRC) approach in which kernel method is used to improve the discrimination
ability of collaborative representation classification (CRC). We then measure
the similarities between the query and atoms in the global dictionary in order
to construct a locality constrained dictionary (LCD) for KCRC. In addition, we
discuss several similarity measure approaches in LCD and further present a
simple yet effective unified similarity measure whose superiority is validated
in experiments. There are several appealing aspects associated with LCD. First,
LCD can be nicely incorporated under the framework of KCRC. The LCD similarity
measure can be kernelized under KCRC, which theoretically links CRC and LCD
under the kernel method. Second, KCRC-LCD becomes more scalable to both the
training set size and the feature dimension. Example shows that KCRC is able to
perfectly classify data with certain distribution, while conventional CRC fails
completely. Comprehensive experiments on many public datasets also show that
KCRC-LCD is a robust discriminative classifier with both excellent performance
and good scalability, being comparable or outperforming many other
state-of-the-art approaches
Explicit MDS Codes with Complementary Duals
In 1964, Massey introduced a class of codes with complementary duals which
are called Linear Complimentary Dual (LCD for short) codes. He showed that LCD
codes have applications in communication system, side-channel attack (SCA) and
so on. LCD codes have been extensively studied in literature. On the other
hand, MDS codes form an optimal family of classical codes which have wide
applications in both theory and practice. The main purpose of this paper is to
give an explicit construction of several classes of LCD MDS codes, using tools
from algebraic function fields. We exemplify this construction and obtain
several classes of explicit LCD MDS codes for the odd characteristic case
Euclidean and Hermitian LCD MDS codes
Linear codes with complementary duals (abbreviated LCD) are linear codes
whose intersection with their dual is trivial. When they are binary, they play
an important role in armoring implementations against side-channel attacks and
fault injection attacks. Non-binary LCD codes in characteristic 2 can be
transformed into binary LCD codes by expansion. On the other hand, being
optimal codes, maximum distance separable codes (abbreviated MDS) have been of
much interest from many researchers due to their theoretical significant and
practical implications. However, little work has been done on LCD MDS codes. In
particular, determining the existence of -ary LCD MDS codes for
various lengths and dimensions is a basic and interesting problem. In
this paper, we firstly study the problem of the existence of -ary
LCD MDS codes and completely solve it for the Euclidean case. More
specifically, we show that for there exists a -ary Euclidean
LCD MDS code, where , or, , and . Secondly, we investigate several constructions of new Euclidean
and Hermitian LCD MDS codes. Our main techniques in constructing Euclidean and
Hermitian LCD MDS codes use some linear codes with small dimension or
codimension, self-orthogonal codes and generalized Reed-Solomon codes
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