Greedy generation of non-binary codes

We get a B-ordering of all binary n-tuples V_n by choosing an ordered basis (y₁, …, y_n) of V _n and ordering the n-tuples as follows: 0, y₁, y ₂, y₂+y₁, y₃, y₃+y₁, y₃+y₂, y₃+y₂+y₁, y₄, … . Given a minimum distance d, choose a set of vectors S with the zero vector first, then go through the vectors in their B-ordering and choose the next vector which has distance d or more from all vectors already chosen. The surprising result that S is linear has been shown in several different ways. Linear codes found in this fashion are called greedy codes. We also look at the non-binary case. One can generalize the concept of B-ordering to the case of an arbitrary base field. We also look at the parity check matrix