In this paper, we consider a method for constructing non-binary self-orthogonal codes from symmetric designs with fixed-point-free automorphisms. All codes over GF(3) and GF(7) derived from symmetric 2-(v,k,λ) designs with fixed-point-free automorphisms of order p for the parameters (v,k,λ,p)=(27,14,7,3),(40,27,18,5) and (45,12,3,5) are classified. A ternary [63,20,21] code with a record breaking minimum weight is constructed from the symmetric 2-(189,48,12) design found recently by Janko. Several codes over GF(5) and GF(7) that are either optimal or have the largest known minimum weight are constructed from designs obtained from known difference sets.
