An \([n,k,d]_q\) code is a linear code of length n, dimension k and minimum weight d over \({\mathbb {F}}_q\) , the field of order q. A fundamental problem in coding theory is to find \(n_q(k,d)\) , the minimum length n for which an \([n,k,d]_q\) code exists for given k, d and q. It is known that the Griesmer bound is attained for all sufficiently large d for fixed q and k. So, a natural question is to find \(D_{q,k}\) , the largest value of d such that the Griesmer bound is not attained for fixed q and k, which is still open for many cases. We pose a conjecture on \(D_{q,k}\) , and we show some cases where our conjecture is valid.
