On the nonexistence of ternary linear codes attaining the Griesmer bound

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 ≥ k with k = 3 , 4 , 5 and q ≥ 2 k - 3 with k ≥ 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 ≤ 9 .