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Báo cáo toán học: "Codes from cubic curves and their extensions"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Codes from cubic curves and their extensions. | Codes from cubic curves and their extensions T. L. Alderson Mathematical Sciences University of New Brunswick Saint John NB E2L 4L5 Canada tim@unbsj.ca A. A. Brueny Electrical and Computer Engineering University of Calgary Calgary AB T2N 1N4 Canada bruen@ucalgary.ca Submitted Aug 13 2007 Accepted Mar 4 2008 Published Mar 12 2008 Mathematics Subject Classification 94B27 Abstract 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 Redei blocking sets and the use of the Bruen-Silverman model of linear codes. 1 Introduction Much of the theory of linear codes is concerned with obtaining bounds on the length of such codes subject to certain constraints involving various parameters such as the minimum distance and with characterization of optimal cases. Similar remarks apply to finite geometries. There one wants to find bounds for example on the maximum or minimum number of points obeying certain combinatorial conditions and the structure in the optimal case. One thinks for example of the famous characterization of conics due The author acknowledges support from the N.S.E.R.C. of Canada yThe author acknowledges support from the N.S.E.R.C. of Canada THE ELECTRONIC JOURNAL OF .