On the Covering Radius of MDS Codes

For a linear maximum distance separable (MDS) code with redundancy r, the covering radius is either r or r -1. However, for r > 3, few examples of q-ary linear MDS codes with radius r -1 are known, including the Reed-Solomon codes with length q + 1. In this paper, for redundancies r as large as 1...

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Bibliographic Details
Published in:IEEE transactions on information theory 2015-02, Vol.61 (2), p.801-811
Main Authors: Bartoli, Daniele, Giulietti, Massimo, Platoni, Irene
Format: Article
Language:English
Subjects:
AG codes
Algebra
Coding theory
covering radius
Electronic mail
Elliptic curves
Error correction & detection
Geometry
MDS codes
Parity check codes
Redundancy
Reed-Solomon codes
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Summary:For a linear maximum distance separable (MDS) code with redundancy r, the covering radius is either r or r -1. However, for r > 3, few examples of q-ary linear MDS codes with radius r -1 are known, including the Reed-Solomon codes with length q + 1. In this paper, for redundancies r as large as 12√q, infinite families of q-ary MDS codes with covering radius r - 1 and length less than q + 1 are constructed. These codes are obtained from algebraic-geometric codes arising from elliptic curves. For most pairs (r, q) with r ≤ 12√q, these are the shortest q-ary MDS codes with covering radius r - 1.
ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2014.2385084