  We construct a ternary [49,25,7] code from the row span of a Jacobsthal
matrix. It is equivalent to a Generalized Quadratic Residue (GQR) code in the
sense of van Lint and MacWilliams (1978). These codes are the abelian
generalizations of the quadratic residue (QR) codes which are cyclic. The union
of the [50,25,8] extension of the said code and its dual supports a
3-(50,14,1248) design. The automorphism group of the latter design is a double
cover of the permutation part of the automorphism group of the [50,25,8] code,
which is isomorphic to $P{\Sigma}L(2,49).$ Other weights in this code, other
GQR codes, and other QR codes yield other 3-designs by the same process. A
simple group action argument is provided to explain this behaviour of isodual
codes.
