An Extension Theorem for Arcs and Linear Codes

We prove the following generalization to the extension theorem of Hill and Lizak: For every nonextendable linear $[n,k,d]_q$ code, $q=p^s$, $(d,q)=1$, we have
$$ \sum_{i\not\equiv0,d\,(\mathrm{mod}\;q)}A_i>q^{k-3}r(q), $$
where $q+r(q)+1$ is the smallest size of a nontrivial blocking set in $\mathrm{PG}(2,q)$. This result is applied further to rule out the existence of some linear codes over $\mathbb F_4$ meeting the Griesmer bound.