Codes from cubic curves and their extensions
The electronic journal of combinatorics, Tome 15 (2008)
We study the linear codes and their extensions associated with sets of points in the plane corresponding to cubic curves. Instead of merely studying linear extensions, all possible extensions of the code are studied. In this way several new results are obtained and some existing results are strengthened. This type of analysis was carried out by Alderson, Bruen, and Silverman [J. Combin. Theory Ser. A, 114(6), 2007] for the case of MDS codes and by the present authors [Des. Codes Cryptogr., 47(1-3), 2008] for a broader range of codes. The methods cast some light on the question as to when a linear code can be extended to a nonlinear code. For example, for $p$ prime, it is shown that a linear $[n,3,n-3]_p$ code corresponding to a non-singular cubic curve comprising $n> p+4$ points admits only extensions that are equivalent to linear codes. The methods involve the theory of Rédei blocking sets and the use of the Bruen-Silverman model of linear codes.
@article{10_37236_766,
author = {T. L. Alderson and A. A. Bruen},
title = {Codes from cubic curves and their extensions},
journal = {The electronic journal of combinatorics},
year = {2008},
volume = {15},
doi = {10.37236/766},
zbl = {1160.94015},
url = {http://geodesic.mathdoc.fr/articles/10.37236/766/}
}
T. L. Alderson; A. A. Bruen. Codes from cubic curves and their extensions. The electronic journal of combinatorics, Tome 15 (2008). doi: 10.37236/766
Cité par Sources :