  In the recent work \cite{shi18}, a combinatorial problem concerning linear
codes over a finite field $\F_q$ was introduced. In that work the authors
studied the weight set of an $[n,k]_q$ linear code, that is the set of non-zero
distinct Hamming weights, showing that its cardinality is upper bounded by
$\frac{q^k-1}{q-1}$. They showed that this bound was sharp in the case $ q=2 $,
and in the case $ k=2 $. They conjectured that the bound is sharp for every
prime power $ q $ and every positive integer $ k $. In this work quickly
establish the truth of this conjecture. We provide two proofs, each employing
different construction techniques. The first relies on the geometric view of
linear codes as systems of projective points. The second approach is purely
algebraic. We establish some lower bounds on the length of codes that satisfy
the conjecture, and the length of the new codes constructed here are discussed.
