21,273 research outputs found
Variety Evasive Sets
We give an explicit construction of a large subset of F^n, where F is a
finite field, that has small intersection with any affine variety of fixed
dimension and bounded degree. Our construction generalizes a recent result of
Dvir and Lovett (STOC 2012) who considered varieties of degree one (affine
subspaces).Comment: 13 page
Subspace Evasive Sets
In this work we describe an explicit, simple, construction of large subsets
of F^n, where F is a finite field, that have small intersection with every
k-dimensional affine subspace. Interest in the explicit construction of such
sets, termed subspace-evasive sets, started in the work of Pudlak and Rodl
(2004) who showed how such constructions over the binary field can be used to
construct explicit Ramsey graphs. More recently, Guruswami (2011) showed that,
over large finite fields (of size polynomial in n), subspace evasive sets can
be used to obtain explicit list-decodable codes with optimal rate and constant
list-size. In this work we construct subspace evasive sets over large fields
and use them to reduce the list size of folded Reed-Solomon codes form poly(n)
to a constant.Comment: 16 page
Linear-algebraic list decoding of folded Reed-Solomon codes
Folded Reed-Solomon codes are an explicit family of codes that achieve the
optimal trade-off between rate and error-correction capability: specifically,
for any \eps > 0, the author and Rudra (2006,08) presented an n^{O(1/\eps)}
time algorithm to list decode appropriate folded RS codes of rate from a
fraction 1-R-\eps of errors. The algorithm is based on multivariate
polynomial interpolation and root-finding over extension fields. It was noted
by Vadhan that interpolating a linear polynomial suffices if one settles for a
smaller decoding radius (but still enough for a statement of the above form).
Here we give a simple linear-algebra based analysis of this variant that
eliminates the need for the computationally expensive root-finding step over
extension fields (and indeed any mention of extension fields). The entire list
decoding algorithm is linear-algebraic, solving one linear system for the
interpolation step, and another linear system to find a small subspace of
candidate solutions. Except for the step of pruning this subspace, the
algorithm can be implemented to run in {\em quadratic} time. The theoretical
drawback of folded RS codes are that both the decoding complexity and proven
worst-case list-size bound are n^{\Omega(1/\eps)}. By combining the above
idea with a pseudorandom subset of all polynomials as messages, we get a Monte
Carlo construction achieving a list size bound of O(1/\eps^2) which is quite
close to the existential O(1/\eps) bound (however, the decoding complexity
remains n^{\Omega(1/\eps)}). Our work highlights that constructing an
explicit {\em subspace-evasive} subset that has small intersection with
low-dimensional subspaces could lead to explicit codes with better
list-decoding guarantees.Comment: 16 pages. Extended abstract in Proc. of IEEE Conference on
Computational Complexity (CCC), 201
Accident Analysis and Prevention: Course Notes 1987/88
This report consists of the notes from a series of lectures given by the authors for a course entitled Accident Analysis and Prevention. The course took place during the second term of a one year Masters degree course in Transport Planning and Engineering run by the Institute for Transport Studies and the Department of Civil Engineering at the University of Leeds. The course consisted of 18 lectures of which 16 are reported on in this document (the remaining two, on Human Factors, are not reported on in this document as no notes were provided). Each lecture represents one chapter of this document, except in two instances where two lectures are covered in one chapter (Chapters 10 and 14). The course first took place in 1988, and at the date of publication has been run for a second time. This report contains the notes for the initial version of the course. A number of changes were made in the content and emphasis of the course during its second run, mainly due to a change of personnel, with different ideas and experiences in the field of accident analysis and prevention. It is likely that each time the course is run, there will be significant changes, but that the notes provided in this document can be considered to contain a number of the core elements of any future version of the course
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