On the nonexistence of ternary linear codes attaining the Griesmer bound

Abstract

An $[n,k,d]_q$ code is a linear code of length n, dimension k and minimum weight d over the field of order q. It is known that the Griesmer bound is attained for all sufficiently large d for fixed q and k. We deal with the problem to find $D_{q,k}$, the largest value of d such that the Griesmer bound is not attained for fixed q and k. $D_{q,k}$ is already known for the cases $q \ge k$ with $k=3,4,5$ and $q \ge 2k-3$ with $k \ge 6$, but not known for the case $q<k$ except for some small q and k. We show that our conjecture on $D_{3,k}$ is valid for $k \le 9$.