Abstract:
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.
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