Summary: It was shown by R. N. Daskalov [Bounds on the minimum length for ternary linear codes of dimension six, Proc. 22nd spring conference of the union of Bulgarian Mathematicians, Sofia, April 2-5, 1993, 15-22 (1994)] that there is no ternary \([30, 6, 18]\) code and \(n_3 (6,18)= 31\), where \(n_3 (k,d)\) denotes the smallest value of \(n\) for which there exists a ternary \([n, k, d]\) code. But it is unknown whether or not there exists a ternary \([29, 6, 17]\) code. The purpose of this paper is to prove the nonexistence of ternary \([29, 6,17]\) codes using a metbod of embedding and finite projective geometries. Since it is known [cf. Table B.4 in N. Hamada and Y. Watamori [Math. Jap. 43, 577-593 (1996; Zbl 0856.94032)] that (i) \(n_3 (6,17)= 29\) or 30 and (ii) \(d_3 (29,6)= 16\) or 17, where \(d_3 (n,k)\) denotes the largest value of \(d\) for which there exists a ternary \([n, k, d]\) code, this implies that \(n_3 (6,17)= 30\) and \(d_3 (29,6)= 16\).