Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Optimal four-dimensional codes over GF(8). | Optimal four-dimensional codes over GF 8 Chris Jones Department of Mathematics and Computer Science St. Mary s College of California Moraga CA 94575 cjones@ Angela Matney Department for the Blind and Vision Impaired 397 Azalea Avenue Richmond VA 23227 Harold Ward Department of Mathematics University of Virginia Charlottesville VA 22904 hnw@ Submitted Oct 21 2005 Accepted Jan 17 2006 Published Apr 28 2006 MR Subject Classifications 94B05 51E22 Abstract We prove the nonexistence of several four-dimensional codes over GF 8 that meet the Griesmer bound. The proofs use geometric methods based on the analysis of the weight structure of subcodes. The specific parameters of the codes ruled out are 111 4 96 110 4 95 102 4 88 101 4 87 93 4 80 and the sequence 29 - j 4 24 - j for j 0 1 2. 1 Introduction An n k d q code is a linear code of length n and dimension k over the finite field GF q for which the minimum distance between different codewords is d. Such a code is traditionally called optimal if n is as small as possible among linear codes with the same k and d. The famous Griesmer bound asserts that the minimum value nq k d of n satisfies n gq k d d d . q d qk-1 and codes meeting this bound are called Griesmer codes. Optimal codes have been the object of research for some time. As with many combinatorial problems dealing with structures meeting bounds optimal codes often exhibit special properties. These generally relate to the geometrical setting for linear codes that is commonly invoked. The THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 R43 1 important theorem of Belov says that if q and k are hxed then Griesmer codes exist for large enough d. Its proof can be framed in a natural way with the geometric setting. The two survey articles by Hill 5 and Hill and Kolev 6 present background material and elaborate on the concepts just described. Hirschfeld s comprehensive book 7 contains a nutshell view of the .
