Weight hierarchies of linear codes satisfying the chain condition (Q1273506)

This paper presents a sufficient condition for the existence of a code satisfying the chain condition with a given weight hierarchy. The weight hierarchy of a linear \([n,k;q]\) code \(C\) over \(\text{GF}(q)\) is the sequence \((d_1, d_2, \ldots, d_k)\) where \(d_r\) is the smallest support of an \(r\)-dimensional subcode of \(C\). \(C\) satisfies the chain condition if there exist subcodes \(D_r\) of \(C\) for \(1 \leq r \leq k\) such that \(\dim(D_r) = r\), \(w_s(D_r) = d_r\) and \(D_1 \subset D_2 \subset \cdots \subset D_k\). \(w_s(D)\) denotes the support weight of \(D\). A sequence \((a_1, a_2, \ldots,a_k)\) of positive integers is chain-good if it is the weight hierarchy of a code satisfying the chain condition. Theorem 1 gives an infinite class of chain-good sequences. This theorem is proved through explicit constructions. Theorem 2 provides additional chain-good sequences.