Design of GuruswamiSudan List Decoding for Elliptic Codes Advancing from ReedSolomon (RS) codes, the length of algebraicgeometric (AG) codes can exceed the size of finite field, resulting in a greater errorcorrection capability. However, this is realized with a genus penalty. Usually, they are not maximum distance separable (MDS) codes. Onepoint elliptic codes are either MDS or almost MDS, yielding a good tradeoff between codeword length and distance property. This paper proposes the GuruswamiSudan (GS) list decoding algorithm for elliptic codes. To define the interpolated polynomial Q(x, y, z), an explicit construction for the zero basis of each affine point is introduced. Given an interpolation multiplicity m, the errorcorrection capability tau_m and the maximum decoding output cardinality l m  of the GS algorithm are characterized. An efficient interpolation algorithm is further presented for elliptic codes. Performance of elliptic codes is shown for the first time, demonstrating their advantage over RS codes.

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.