An extension theorem for arcs and linear codes

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 $$\sum\limits_{i\not \equiv 0,d(\bmod q)} {A_i > q^{k - 3} r(q),} $$ 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.