Dual MacWilliams pair

A pair of posets (P,Q ) on [n] is called a weak dual MacWilliams pair (wdMp) if the P-weight enumerator of a linear code uniquely determines the Q-weight enumerator of the dual of that code for every linear code of length n over a finite field. First, we show that (P, Pbreve) is a wdMp if and only if the group of all P-weight preserving linear automorphisms of the ambient n-dimensional space over the finite field acts transitively on every P-sphere centered at 0. Here Pbreve is the dual poset of P. Also, we show some equivalent conditions which say that P being weak order poset with Q = Pbreve is essentially the only possible case for (P,Q) to be a wdMp