On linear lexicographic codes: Ninth column construction of the ternary Golay code

We characterize linear lexicographic $p$-ary codes. Using this characterization, when $p\ge 3$, we determine the dimensions of linear lexicographic codes obtained from several bases including the standard basis, except for those of certain minimum distances. In these excluded cases, we may obtain linear codes of higher dimensions; for instance, when $p=3$ and $d=6$, the ternary Golay code is obtained.