The authors derive a number of bounds that generalize known bounds of linear codes involving: (1) \(n\), the length of a linear code \(C\), (2) \(k\), the dimension of a linear code over \(\text{GF} (q)\), (3) \(d_r\), the \(r\)-th minimum support weight. A number of bounds on \(d_r (C)\) are derived, generalizing the Plotkin bound and the Griesmer bound, as well as giving two new existential bounds. The main result is that there exist codes of any given rate \(R\) whose ratio \(d_r/ d_1\) is lower bounded by a number ranging from \((q^r-1)/ (q^r- q^{r-1})\) to \(r\), depending on \(R\).