On the second greedy weight for linear codes of dimension 3

The difference g₂−d₂ for a q-ary linear n,3,d] code C is studied. Here d₂ is the second generalized Hamming weight, that is, the smallest size of the support of a 2-dimensional subcode of C; and g₂ is the second greedy weight, that is, the smallest size of the support of a 2-dimensional subcode of C which contains a codeword of weight d. For codes of dimension 3, it is shown that the problem is essentially equivalent to finding certain weighting of the points in the projective plane, and weighting which give the maximal value of g₂−d₂ are determined in almost all cases. In particular max(g₂−d₂) is determined in all cases for q⩽9.