Abstract
To determine all possible weight hierarchies of general linear codes is a basic theory having important scientific significance raised in the communication system. But when k or q is larger, it is impossible for q-ary linear codes of dimension k. Reasonable formulation of the problem is modified: to determine almost all the weight hierarchies of q-ary general linear codes of dimension k. Based on the finite projective geometry method, q-ary linear codes of dimension 5 in class V are studied in this paper, in which we find new necessary conditions of weight hierarchies of 5-dimensional codes in class V, and classify the weight hierarchies of 5-dimensional codes in class V into two subclass,and advance subspace set method, and determine almost all the weight hierarchies of q-ary linear codes of dimension 5 in class V. It open the way for the remaining three class of weight hierarchies of 5-dimensional codes, and break through the difficulty. Furthermore, new necessary conditions show that original necessary conditions of weight hierarchies are not enough. We need to develop further new necessary conditions, in order to attack and solve problems of dimension k.
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