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\limits_{i\not \equiv 0,d(\bmod q)} {A_i > q^{k - 3} r(q),} $ where q + r(q) + 1 is the smallest size of a nontrivial blocking set in 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.