Given a prime power *q*, and positive integers *n*, *k*, and *d*, one of the classical problems of algebraic coding theory is to ask whether there exists a *k*-dimensional subspace of the vector space of *n*-tuples over the Galois field *GF*(*q*) having minimal Hamming distance *d*. (Such a space is called an [*n*,*k*,*d*]-code over *GF*(*q*).) In particular, given *k*, *d*, and *q*, one can ask, “What is the minimal value of *n* for which there is a positive answer to this equation?” A lower bound for such a value, known as the Griesmer bound, is established in any elementary book on algebraic coding theory. However, whether or not the Griesmer bound is in fact attainable for a given specific case is not known in general.

In this paper, the authors use sophisticated mathematical techniques to consider some special cases of this problem. They show the nonexistence of certain codes: there does not exist a [109,4,86]-code over *GF*(5), a [238,5,177]-code over *GF*(4), or a [478,4,417]-code over *GF*(8). Moreover, the authors also establish conditions for the Griesmer bound, or the Griesmer bound + 1, to be attainable. While highly technical, this paper has important applications in communication theory.