Codes from Cubic Curves and their Extensions 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(13), 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,n3_p code corresponding to a nonsingular cubic curve comprising nu003e p4 points admits only extensions that are equivalent to linear codes.  The methods involve the theory of Redei blocking sets and the use of the BruenSilverman model of linear codes.

Asymptotically good ZprZpsadditive cyclic codes Abstract   We construct a class of      Z      p    r        Z      p    s       additive cyclic codes generated by pairs of polynomials, where p is a prime number. Based on probabilistic arguments, we determine the asymptotic rates and relative distances of this class of codes: the asymptotic GilbertVarshamov bound at      1      p    s  xe2x88x92  r      2    xcexb4    is greater than      1    2      and the relative distance of the code is convergent to xcexb4, while the rate is convergent to      1    1      p    s  xe2x88x92  r        for    0    xcexb4      1    1      p    s  xe2x88x92  r        and    1  xe2x89xa4  r    s   . As a consequence, we prove that there exist numerous asymptotically good      Z      p    r        Z      p    s       additive cyclic codes.