Weight hierarchies of linear codes satisfying the almost chain condition

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 size of the smallest support of an r-dimensional subcode of C. An [n, k; q] code satisfies the chain condition if there exists subcodes D-1 subset of D-2 subset of ... subset of 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 of D-3 subset of ... subset of D-k = C and D-1 subset of 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.