Weight hierarchies of linear codes satisfying the almost chain condition (Q866019)

The weight hierarchy of a linear \([n,k;q]\) code \(C\) over \(\mathrm{GF}(q)\) is the sequence \((d_1,d_2,\dots,d_k)\) where \(d_r\) is the size of the smallest support of an \(r\)-dimensional subcode of \(C\). An \([n,k;q]\) code satisfies the chain condition if there exist subcodes \(D_1\subset D_2\subset\dots\subset D_k=C\) of \(C\) such that \(D_r\) has dimension \(r\) and support of size \(d_r\) for all \(r\). Further, \(C\) satisfies the almost chain condition if it does not satisfy the chain condition, but there exist subcodes \(D_r\) of dimension \(r\) and support of size \(d_r\) for all \(r\) such that \(D_2\subset D_3\subset\dots\subset D_k=C\) and \(D_1\subset D_3\). A simple necessary condition for a sequence to be the weight hierarchy of a code satisfying the almost chain condition is given. Further, explicit constructions of such codes are given, showing that in almost all cases, the necessary conditions are also sufficient.