The Determination on Weight Hierarchies of q-Ary Linear Codes of Dimension 5 in Class IV

• Published in: Journal of systems science and complexity 2016-02, Vol.29 (1), p.243-258
• Main Authors: Wang, Lijun, Chen, Wende
• Format: Article
• Language: English
• Subjects: Complex Systems; Control; Mathematics; Mathematics and Statistics; Mathematics of Computing; Operations Research/Decision Theory; Statistics; Systems Theory; 理论; 系统学; 系统工程; 系统科学
• ISSN: 1009-6124; 1559-7067
• DOI: 10.1007/s11424-015-4072-6

The weight hierarchy of a linear[n;k;q]code C over GF（q） is the sequence（d_1,d_2,…,d_k）where d_r is the smallest support of any r-dimensional subcode of C. ＂Determining all possible weight hierarchies of general linear codes＂ is a basic theoretical issue and has important scientific significance in communication system.However,it is impossible for g-ary linear codes of dimension k when q and k are slightly larger,then a reasonable formulation of the problem is modified as： ＂Determine almost all weight hierarchies of general g-ary linear codes of dimension k＂.In this paper,based on the finite projective geometry method,the authors study g-ary linear codes of dimension 5 in class IV,and find new necessary conditions of their weight hierarchies,and classify their weight hierarchies into6 subclasses.The authors also develop and improve the method of the subspace set,thus determine almost all weight hierarchies of 5-dimensional linear codes in class IV.It opens the way to determine the weight hierarchies of the rest two of 5-dimensional codes（classes III and VI）,and break through the difficulties.Furthermore,the new necessary conditions show that original necessary conditions of the weight hierarchies of k-dimensional codes were not enough（not most tight nor best）,so,it is important to excogitate further new necessary conditions for attacking and solving the fc-dimensional problem.