20,661 research outputs found
On the minimum distance of elliptic curve codes
Computing the minimum distance of a linear code is one of the fundamental
problems in algorithmic coding theory. Vardy [14] showed that it is an \np-hard
problem for general linear codes. In practice, one often uses codes with
additional mathematical structure, such as AG codes. For AG codes of genus
(generalized Reed-Solomon codes), the minimum distance has a simple explicit
formula. An interesting result of Cheng [3] says that the minimum distance
problem is already \np-hard (under \rp-reduction) for general elliptic curve
codes (ECAG codes, or AG codes of genus ). In this paper, we show that the
minimum distance of ECAG codes also has a simple explicit formula if the
evaluation set is suitably large (at least of the group order). Our
method is purely combinatorial and based on a new sieving technique from the
first two authors [8]. This method also proves a significantly stronger version
of the MDS (maximum distance separable) conjecture for ECAG codes.Comment: 13 page
Algebraic geometric codes
The performance characteristics are discussed of certain algebraic geometric codes. Algebraic geometric codes have good minimum distance properties. On many channels they outperform other comparable block codes; therefore, one would expect them eventually to replace some of the block codes used in communications systems. It is suggested that it is unlikely that they will become useful substitutes for the Reed-Solomon codes used by the Deep Space Network in the near future. However, they may be applicable to systems where the signal to noise ratio is sufficiently high so that block codes would be more suitable than convolutional or concatenated codes
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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