291,168 research outputs found
Decoding Cyclic Codes up to a New Bound on the Minimum Distance
A new lower bound on the minimum distance of q-ary cyclic codes is proposed.
This bound improves upon the Bose-Chaudhuri-Hocquenghem (BCH) bound and, for
some codes, upon the Hartmann-Tzeng (HT) bound. Several Boston bounds are
special cases of our bound. For some classes of codes the bound on the minimum
distance is refined. Furthermore, a quadratic-time decoding algorithm up to
this new bound is developed. The determination of the error locations is based
on the Euclidean Algorithm and a modified Chien search. The error evaluation is
done by solving a generalization of Forney's formula
Decoding Reed-Muller codes over product sets
We give a polynomial time algorithm to decode multivariate polynomial codes
of degree up to half their minimum distance, when the evaluation points are
an arbitrary product set , for every . Previously known
algorithms can achieve this only if the set has some very special algebraic
structure, or if the degree is significantly smaller than . We also
give a near-linear time randomized algorithm, which is based on tools from
list-decoding, to decode these codes from nearly half their minimum distance,
provided .
Our result gives an -dimensional generalization of the well known decoding
algorithms for Reed-Solomon codes, and can be viewed as giving an algorithmic
version of the Schwartz-Zippel lemma.Comment: 25 pages, 0 figure
Toric surface codes and Minkowski sums
Toric codes are evaluation codes obtained from an integral convex polytope and finite field \F_q. They are, in a sense, a natural
extension of Reed-Solomon codes, and have been studied recently by J. Hansen
and D. Joyner. In this paper, we obtain upper and lower bounds on the minimum
distance of a toric code constructed from a polygon by
examining Minkowski sum decompositions of subpolygons of . Our results give
a simple and unifying explanation of bounds of Hansen and empirical results of
Joyner; they also apply to previously unknown cases.Comment: 15 pages, 7 figures; This version contains some minor editorial
revisions -- to appear SIAM Journal on Discrete Mathematic
Construction of Rational Surfaces Yielding Good Codes
In the present article, we consider Algebraic Geometry codes on some rational
surfaces. The estimate of the minimum distance is translated into a point
counting problem on plane curves. This problem is solved by applying the upper
bound "\`a la Weil" of Aubry and Perret together with the bound of Homma and
Kim for plane curves. The parameters of several codes from rational surfaces
are computed. Among them, the codes defined by the evaluation of forms of
degree 3 on an elliptic quadric are studied. As far as we know, such codes have
never been treated before. Two other rational surfaces are studied and very
good codes are found on them. In particular, a [57,12,34] code over
and a [91,18,53] code over are discovered, these
codes beat the best known codes up to now.Comment: 20 pages, 7 figure
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