The determination on weight hierarchies of q-ary linear codes of dimension 5 in class IV

Abstract

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 q-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 q-ary linear codes of dimension k”. In this paper, based on the finite projective geometry method, the authors study q-ary linear codes of dimension 5 in class IV, and find new necessary conditions of their weight hierarchies, and classify their weight hierarchies into 6 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 k-dimensional problem.