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{\bf T. L. Alderson and A. A. Bruen}
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{\bf Codes from Cubic Curves and their Extensions}
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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 [{\em J. Combin. Theory Ser. A},
114(6), 2007] for the case of MDS codes and by the present authors
[{\em 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 R\'edei blocking sets and the use of the Bruen-Silverman
model of linear codes.



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