6 research outputs found
Maximum Distance Separable Codes and Arcs in Projective Spaces
Given any linear code over a finite field we show how can be
described in a transparent and geometrical way by using the associated
Bruen-Silverman code. Then, specializing to the case of MDS codes we use our
new approach to offer improvements to the main results currently available
concerning MDS extensions of linear MDS codes. We also sharply limit the
possibilities for constructing long non-linear MDS codes.Comment: 18 Pages; co-author added; some results updated; references adde
Cubic Curves, Finite Geometry and Cryptography
Some geometry on non-singular cubic curves, mainly over finite fields, is
surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are
classified accordingly. The group structure and the possible numbers of
rational points are also surveyed. A possible strengthening of the security of
elliptic curve cryptography is proposed using a `shared secret' related to the
group law. Cubic curves are also used in a new way to construct sets of points
having various combinatorial and geometric properties that are of particular
interest in finite Desarguesian planes.Comment: This is a version of our article to appear in Acta Applicandae
Mathematicae. In this version, we have corrected a sentence in the third
paragraph. The final publication is available at springerlink.com at
http://www.springerlink.com/content/xh85647871215644
Near MDS poset codes and distributions
We study -ary codes with distance defined by a partial order of the
coordinates of the codewords. Maximum Distance Separable (MDS) codes in the
poset metric have been studied in a number of earlier works. We consider codes
that are close to MDS codes by the value of their minimum distance. For such
codes, we determine their weight distribution, and in the particular case of
the "ordered metric" characterize distributions of points in the unit cube
defined by the codes. We also give some constructions of codes in the ordered
Hamming space.Comment: 13 pages, 1 figur
A characterization of MDS codes that have an error correcting pair
Error-correcting pairs were introduced in 1988 by R. Pellikaan, and were
found independently by R. K\"otter (1992), as a general algebraic method of
decoding linear codes. These pairs exist for several classes of codes. However
little or no study has been made for characterizing those codes. This article
is an attempt to fill the vacuum left by the literature concerning this
subject. Since every linear code is contained in an MDS code of the same
minimum distance over some finite field extension we have focused our study on
the class of MDS codes.
Our main result states that an MDS code of minimum distance has a
-ECP if and only if it is a generalized Reed-Solomon code. A second proof is
given using recent results Mirandola and Z\'emor (2015) on the Schur product of
codes