tailieunhanh - Báo cáo toán học: "Multi-covering Radius for Rank Metric Codes"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Multi-covering Radius for Rank Metric Codes. | Multi-covering Radius for Rank Metric Codes W. B. Vasantha R. S. Selvaraj Department of Mathematics Department of Mathematics Indian Institute of Technology Madras National Institute of Technology Warangal Chennai - 600 036 India Warangal - 506 004 India vasantha@ rsselva@ Submitted Dec 10 2008 Accepted Nov 29 2009 Published Dec 8 2009 Mathematics Subject Classifications 94B65 94B75 05B40 11H31 15A03 Abstract The results of this paper are concerned with the multi-covering radius a generalization of covering radius of Rank Distance RD codes. This leads to greater understanding of RD codes and their distance properties. Results on multi-covering radii of RD codes under various constructions are given by varying the parameters. Some bounds are established. A relationship between multi-covering radii of an RD code and that of its ambient space is also found. The classical sphere bound is generalized. 1 Introduction The concept of covering radius has been the subject of hundreds of papers. 2 3 can be referred for a comprehensive survey and thorough bibliography on the subject. In this paper simultaneous coverings of m-tuples of vectors rather than single vector are investigated for codes over the Galois field F2N defined with rank metric. The notion of multi-covering radius a generalization of the covering radius was introduced by Andrew Klapper 8 for binary codes with Hamming metric to study the existence of stream ciphers secured against a large class of attacks. Here for the first time study of multi-covering radius for codes with a non-Hamming metric namely rank metric is carried out. Recall that an RD code 5 of length n is a subset of F v where n N and N 1 q being a power of a prime wherein the weight rank norm of each vector is defined to be the maximum number of its coordinates that are linearly independent and the corresponding metric induced by this norm is called Thanks to Council of Scientific and Industrial Research CSIR India for its .